a constrained-total-least-squares method for joint...

Research ArticleA Constrained-Total-Least-Squares Method for Joint Estimationof Source and Sensor Locations A General Framework

DingWang 12 Ruirui Liu 12 Jiexin Yin12 ZhidongWu12

YunlongWang12 and ChengWang12

1National Digital Switching System Engineering and Technology Research Center Zhengzhou 450002 China2Zhengzhou Information Science and Technology Institute Zhengzhou Henan 450002 China

Correspondence should be addressed to Ruirui Liu liu rr927163com

Received 14 July 2017 Accepted 8 February 2018 Published 5 April 2018

Academic Editor Filippo Ubertini

Copyright copy 2018 Ding Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

It is well known that sensor location uncertainties can significantly deteriorate the source positioning accuracy Thereforeimproving the sensor locations is necessary in order to achieve better localization performance In this paper a constrained-total-least-squares (CTLS) method for simultaneously locating multiple disjoint sources and refining the erroneous sensor positions ispresented Unlike conventional localization techniques an important feature of the proposed method is that it establishes a generalframework that is suitable for many different location measurements First a modified CTLS optimization problem is formulatedafter some algebraicmanipulations and then the correspondingNewton iterative algorithm is derived to give the numerical solutionSubsequently by using the first-order perturbation analysis the explicit expression for the covariance matrix of the proposed CTLSestimator is deduced under the Gaussian assumption Moreover the estimation accuracy of the CTLS method is shown to achievethe Cramer-Rao bound (CRB) before the thresholding effect occurs by a rigorous proof Finally two kinds of numerical examplesare given to corroborate the theoretical development in this paper One uses the TDOAsGROAs measurements and the other isbased on the TOAsFOAs parameters

1 Introduction

Passive source localization has attracted significant attentionin the signal processing research due to its importance tomany applications including radar sonar microphone arraysnavigation sensor networks and wireless communicationsCommon wireless location systems are based on a two-step procedure for target position determination In the firstphase the intermediate parameters that depend on the loca-tions of the sources are estimated from the received signals Ingeneral these parameters include direction of arrival (DOA)[1ndash3] time of arrival (TOA) [4ndash7] time difference of arrival(TDOA) [8ndash17] frequency of arrival (FOA) [18] frequencydifference of arrival (FDOA) [19ndash28] received signal strength(RSS) [29] and gain ratios of arrival (GROA) [30 31] In thesecond phase the previously estimated parameters are usedto locate the sources During the past few decades numerousmethods have been proposed for the two active research

areas In this paper we focus on the latter that is emitterlocation estimation

It is easy to see that the position determination isequivalent to solving a set of nonlinear equations relating theintermediate parameters to the coordinates of the sources Anumber of localizationmethods are available in the literatureSome of them are iterative algorithms (such as Taylor series(TS) algorithm [2 12 20 23 32] and constrained-total-least-squares (CTLS) algorithm [3 6 15 17 25ndash27]) that requireproper initial solution guesses and the others are closed-formsolutions (such as total least squares (TLS) solution [1 11 22]quadratic constraint least square (QCLS) solution [4 9 1013 33] and two-step weighted least square (TWLS) solution[5 7 8 14 16 19 21 24 28ndash31]) that aremore computationallyefficient Most of these algorithms can reach the correspond-ing Cramer-Rao bound (CRB) accuracy under moderatelevel of signal-to-noise ratio (SNR) Moreover it is worthnoting that all the localization algorithms need to transform

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 8475693 23 pageshttpsdoiorg10115520188475693

2 Mathematical Problems in Engineering

the nonlinear measurement equations into pseudo-linearequations except for the TS algorithm

However it must be emphasized that the accuracy ofsource location estimate may be seriously degraded by thesensor location errors regardless of the specific localizationalgorithm used In [21 34] the source location mean squareerror (MSE) is derivedwhen the sensor locations are assumedcorrect but in fact have errors Generally there exist twoclasses of methods that can mitigate the effects of theuncertainties in sensor location The first class of methods isto incorporate the statistical characteristic of sensor locationerrors into the position estimation procedure [5 6 17 21 2426 27] and the second one performs joint estimation of theunknown source locations and the inaccurate sensor posi-tions together [2 7 12 14 16 20 23 28 31] In this work weconcentrate on the latter because it can increase the accuracyof the sensor position estimates and tolerate higher noise levelbefore the thresholding effect caused by nonlinear estimationstarts to occur

It is well known that the TLS technique is an improvedleast squares (LS) method to solve an overdetermined setof linear equations Ax asymp y when there are errors not onlyin the observations y but in the coefficient matrix A aswell In [1 11 22] the TLS method is applied to sourcelocalization Note that the TLS solution can be found simplyvia the singular value decomposition (SVD) technique [35]and therefore it is computationally attractive Howeverthe TLS estimator is generally not asymptotically efficientbecause it assumes that the noise components in A and yare independent and identically distributed (iid) which israrely realistic in practical scenario The CTLS method as anatural extension of the TLS method is able to fully exploitthe noise structure in A and y [36] and hence the resultingsolution is shown to achieve the Cramer-Rao bound (CRB)Indeed the CTLS method has been successfully applied towireless location In [3] the CTLS algorithm is proposedto solve the bearing-only localization problem In [15] theCTLS localization algorithm using TDOA measurementsis presented Additionally an efficient CTLS algorithm fordetermining the position and velocity of a moving sourcebased on TDOA and FDOA measurements is developed in[25] However it is noteworthy that none of these CTLSalgorithms consider the sensor position errors which mayseriously deteriorate the positioning accuracy In order toreduce the effects of the uncertainties in sensor positions therobust CTLS localization algorithms are presented in [6 2627]

While the above-mentioned CTLS algorithms canachieve satisfactory performance it is necessary to pointout that all of them apply only to the single-source scenarioand moreover none of them can provide the joint estimationof source and sensor locations Furthermore all thealgorithm derivations and theoretical analysis are performedonly for some specific measurements thus leading tothe lack of a united framework for this problem Thispaper presents an efficient CTLS method that can locatemultiple disjoint sources and refine the erroneous sensorpositions simultaneously Different from the existingapproaches the proposed method is derived in a more

general framework that is applicable to many differentlocation measurements First a modified CTLS optimizationproblem is formulated after some algebraic manipulationsand then the corresponding Newton iterative algorithmis derived to yield the numerical solution Subsequentlyby exploiting the first-order perturbation analysis theclosed-form expression for the covariance matrix of the newCTLS estimator is deduced under the Gaussian assumptionMoreover the estimation accuracy of the CTLS solution isrigorously proved to reach the CRB before the thresholdingeffect occurs Finally we give two examples to illustrate howto utilize the proposed CTLS method for source localizationOne uses the TDOAsGROAsmeasurements and the other isbased on the TOAsFOAs parametersThe experiment resultssupport the theoretical development in this paper

The remainder of this paper is organized as follows InSection 2 the measurement model for source localization isdescribed and the problem under investigation is formulatedSection 3 derives the modified CTLS optimization modelIn Section 4 the Newton iterative algorithm is derived toprovide the joint estimation of source and sensor loca-tions Section 5 provides the closed-form expression for thecovariance matrix of the new CTLS estimator and provesits asymptotical efficiency In Section 6 two examples aregiven to illustrate how to utilize the proposed CTLS methodfor source localization Numerical simulations are presentedin Section 7 to support the theoretical development in thispaper Conclusions are drawn in Section 8 The proofs of themain results are shown in the Appendixes (available here)

2 Measurement Model andProblem Formulation

21 Nonlinear Measurement Model We consider the local-ization scenario where 119863 disjoint sources are to be locatedUnder ideal condition the location observation equationassociated with the 119889th source can be represented in a genericform as

z푑0 = f (u푑w) (1 le 119889 le 119863) (1)

where z푑0 isin R푝1times1 is the true measurement vector u푑 isinR푝2times1 is the position andor velocity vector of the 119889th sourcew isin R푝3times1 denotes the system parameter which contains thesensor positions andor velocities and f(sdot sdot) is the nonlinearfunction that depends on the specificmeasurement type used

Note that if the vectors z푑0 and w can be accuratelyobtained the localization problem is equivalent to solvinga set of nonlinear equations However z푑0 and w are notknown exactly in practice First only the noisy version of z푑0denoted as z푑 is available It can be written as

z푑 = z푑0 + n푑 = f (u푑w) + n푑 (1 le 119889 le 119863) (2)

where n푑 is the measurement noise vector that follows zero-mean Gaussian distribution with covariance matrix N푑 =119864[n푑n푇푑 ] In addition the known system parameter denotedas k is also erroneous It can be modeled as

k = w +m (3)

Mathematical Problems in Engineering 3

wherem is the noise vector and it is Gaussian distributedwithzero-mean and covariance matrixM = 119864[mm푇] Besidesmand n푑1le푑le퐷 are statistically independent22 Pseudo-Linear Measurement Model For some specialmeasurements (eg DOA TOA TDOA and GROA) (1) canbe transformed into the following pseudo-linear model

a (z푑0w) = B (z푑0w) t푑 = B (z푑0w) h (u푑w)(1 le 119889 le 119863) (4)

where a(z푑0w) isin R푝1times1 is the pseudo-linear measurementvector and B(z푑0w) isin R푝1times(푝2+푝4) is the coefficient matrixt푑 = h(u푑w) isin R(푝2+푝4)times1 Vector function h(u푑w) is givenby

h (u푑w) = [[u푑 minus Jw

s (u푑w) ]] (5)

where J isin R푝2times푝3 is a known and constant matrix and s(u푑w) isin R푝4times1 comprises all the instrumental variables whosenumber is defined by 1199014

Since every equation in (4) is related to the systemparameter w we must combine these equations to performjoint estimation of all the position vectors u푑1le푑le퐷 andthe system parameter w In this treatment we can obtaincooperation gain compared to the approaches which locatethe sources individually

Putting all the119863 equations in (4) together yields

a (z0w) = B (z0w) t = B (z0w) h (uw) (6)

where

a (z0w)= [(a (z10w))푇 (a (z20w))푇 sdot sdot sdot (a (z퐷0w))푇]푇isin R푝1퐷times1

B (z0w)= blkdiag [B (z10w) B (z20w) sdot sdot sdot B (z퐷0w)]isin R푝1퐷times(푝2+푝4)퐷

t = h (uw) = [t푇1 t푇2 sdot sdot sdot t푇퐷]푇= [(h (u1w))푇 (h (u2w))푇 sdot sdot sdot (h (u퐷w))푇]푇isin R(푝2+푝4)퐷times1

z0 = [z푇10 z푇20 sdot sdot sdot z푇퐷0]푇 isin R푝1퐷times1u = [u푇1 u푇2 sdot sdot sdot u푇퐷]푇 isin R푝2퐷times1

(7)

It is obvious from (7) that vector u contains the locationvectors of all the emitters In addition it can also been seen

from (7) that vector z0 comprises the measurement vectors ofall the sources The noisy version of z0 is denoted by z whichcan be expressed as

z = z0 + n

= [(f (u1w))푇 (f (u2w))푇 sdot sdot sdot (f (u퐷w))푇]푇+ n = f (uw) + n

(8)

where

n = [n푇1 n푇2 sdot sdot sdot n푇퐷]푇 isin R푝1퐷times1

f (uw)= [(f (u1w))푇 (f (u2w))푇 sdot sdot sdot (f (u퐷w))푇]푇isin R푝1퐷times1

(9)

It can be readily seen from (9) that the noise vectorn follows zero-mean Gaussian distribution Its covariancematrix is defined by N = 119864[nn푇] If n푑1 and n푑2 arestatistically independent for 1198891 = 1198892 then we have N =blkdiag [N1 N2 sdot sdot sdot N퐷]

The positioning problem here can be briefly stated asfollowsGiven the observation vectors z푑1le푑le퐷 and availablesystem parameter k find an estimate of u푑1le푑le퐷 (or u) andw as accurate as possible based on the pseudo-linear equation(6)

3 Optimization Model

In (6) the functional forms of a(sdot sdot) and B(sdot sdot) are knownbut vectors z0 and w are not available and only their noisyvalues (ie z and k) can be obtained In order to establishthe CTLS optimization model we shall perform a first-orderTaylor series expansion of a(z0w) and B(z0w) around z aswell as k It can be verified that

a (z0w) asymp a (z k) minus A1 (z k) n minus A2 (z k)mB (z0w) asymp B (z k) minus 푝1퐷sum

푗=1

⟨n⟩푗 sdot B1푗 (z k) minus 푝3sum푗=1

⟨m⟩푗sdot B2푗 (z k)

(10)

where

A1 (z k) = 120597a (z k)120597z푇 isin R푝1퐷times푝1퐷A2 (z k) = 120597a (z k)120597k푇 isin R푝1퐷times푝3

B1푗 (z k) = 120597B (z k)120597 ⟨z⟩푗 isin R푝1퐷times(푝2+푝4)퐷 (1 le 119895 le 1199011119863)B2푗 (z k) = 120597B (z k)120597 ⟨k⟩푗 isin R푝1퐷times(푝2+푝4)퐷 (1 le 119895 le 1199013)

(11)


Inserting (10) into (6) leads to

a (z k) minus A1 (z k) n minus A2 (z k)masymp B (z k) t minus 푝1퐷sum

푗=1

⟨n⟩푗 sdot B1푗 (z k) tminus 푝3sum푗=1

⟨m⟩푗 sdot B2푗 (z k) t 997904rArra (z k) minus B (z k) t asymp C1 (t z k) n + C2 (t z k)m

(12)

where

C1 (t z k)= A1 (z k)minus [ B11 (z k) t B12 (z k) t sdot sdot sdot B1푝1퐷 (z k) t]

isin R푝1퐷times푝1퐷C2 (t z k)

= A2 (z k)minus [ B21 (z k) t B22 (z k) t sdot sdot sdot B2푝3 (z k) t]

isin R푝1퐷times푝3

(13)

Note that the problem addressed herein is the jointestimation of u and w Therefore it is necessary to define anaugmented parameter vector as below

t = h (uw) = [ tw] = [h (uw)

w] isin R((푝2+푝4)퐷+푝3)times1 (14)

Then by combining (3) and (12) we can get the followingprogramming model

minuwnm

[ nm]푇 sdot [ Nminus1 O푝1퐷times푝3

O푝3times푝1퐷 Mminus1] sdot [ n

m]

st [a (z k)k

] minus [ B (z k) O푝1퐷times푝3O푝3times(푝2+푝4)퐷 I푝3

] sdot t= [C1 (t z k) C2 (t z k)

O푝3times푝1퐷 I푝3] sdot [ n

m]

(15)

Although (15) has equality constraint it can be converted intoan unconstrained minimization problem over u and w Thedetails can be found in the following proposition

Proposition 1 If C1(t z k) is an invertible matrix then theconstrained optimization problem (15) can be recast as anequivalent unconstrained one which is expressed as

minuw

119869ctls (uw) = minuw

(a (z k) minus B (z k) t)푇 (Q (t z k))minus1 (a (z k) minus B (z k) t) (16)

where

a (z k) = [a (z k)k

] isin R(푝1퐷+푝3)times1B (z k) = [ B (z k) O푝1퐷times푝3

O푝3times(푝2+푝4)퐷 I푝3] isin R(푝1퐷+푝3)times((푝2+푝4)퐷+푝3)

(17)

Q (t z k) = [[[[C1 (t z k) N (C1 (t z k))푇 + C2 (t z k)M (C2 (t z k))푇 C2 (t z k)M

M (C2 (t z k))푇 M

]]]]isin R(푝1퐷+푝3)times(푝1퐷+푝3) (18)

Proof Define n耠 = Nminus12n and m耠 = Mminus12m and then (15)is equivalent to

minuwn1015840 m1015840

10038171003817100381710038171003817100381710038171003817100381710038171003817[n耠

m耠]100381710038171003817100381710038171003817100381710038171003817100381710038172

2

st [a (z k)k


] sdot t= [C1 (t z k) N12 C2 (t z k)M12

O푝3times푝1퐷 M12]


sdot [ n耠m耠

](19)

The optimal solution to (19) is given by

[ n耠m1015840

]opt

= [C1 (t z k) N12 C2 (t z k)M12O푝3times푝1퐷 M12

]dagger

sdot ([a (z k)k


] sdot t)= [C1 (t z k) N12 C2 (t z k)M12

O푝3times푝1퐷 M12]dagger

sdot (a (z k) minus B (z k) t)

(20)

where (sdot)dagger represents the Moore-Penrose inverseSince C1(t z k) is invertible it can be checked that[ C1 (tzk)N12 C2 (tzk)M12

O1199013times1199011119863 M12 ] has full row rank which leadsto

([C1 (t z k) N12 C2 (t z k)M12O푝3times푝1퐷 M12

]dagger)푇

sdot [C1 (t z k) N12 C2 (t z k)M12O푝3times푝1퐷 M12

]dagger

= ([C1 (t z k) N12 C2 (t z k)M12O푝3times푝1퐷 M12

]


]푇)minus1

= (Q (t z k))minus1

(21)

Combining (20) and (21) yields

10038171003817100381710038171003817100381710038171003817100381710038171003817[n耠

m耠]opt

100381710038171003817100381710038171003817100381710038171003817100381710038172

2

= (a (z k) minus B (z k) t)푇 (Q (t z k))minus1

sdot (a (z k) minus B (z k) t)(22)

which combined with (19) proves the proposition

We would like to emphasize that (16) is the CTLSoptimization model to jointly estimate source position u

and system parameter w simultaneously Moreover it is ageneric model that can be applied to many different locationmeasurements In the next section the numerical algorithmto solve (16) is derived

4 Numerical Algorithm

It is obvious that (16) is a nonlinear minimization problemTherefore the analytical solution is in general not availableand a numerical technique is required One widely appliednumerical method is Newton iteration which has two-orderconvergence rate if the function to be minimized is twicedifferentiable Note that in each iteration step the gradientandHessianmatrix of the object functionmust be computedHence we need to derive the explicit expressions for thegradient and Hessian matrix

For notational convenience the cost function 119869ctls(uw)in (16) is rewritten as

119869ctls (uw) = (g (uw))푇G (uw) g (uw) (23)

where

g (uw) = B (z k) t minus a (z k) G (uw) = (Q (t z k))minus1 (24)

From (23) the gradient of 119869ctls(uw) can be expressed as

120593 (uw) = [[[[[[

120597119869ctls (uw)120597u120597119869ctls (uw)120597w]]]]]]= 1205931 (uw) + 1205932 (uw) (25)

where

1205931 (uw) = [[[[[[2(120597g (uw)120597u푇 )푇G (uw) g (uw)2 (120597g (uw)120597w푇 )푇G (uw) g (uw)

]]]]]]1205932 (uw)

= [[[[[[(120597vec (G (uw))120597u푇 )푇 (g (uw) otimes g (uw))(120597vec (G (uw))120597w푇 )푇 (g (uw) otimes g (uw))

]]]]]]

(26)

Applying (25) the Hessian matrix of 119869ctls(uw) is given by

Ψ (uw) = [ 120597120593 (uw)120597u푇 120597120593 (uw)120597w푇 ]

= [[[[1205972119869ctls (uw)120597u120597u푇 1205972119869ctls (uw)120597u120597w푇1205972119869ctls (uw)120597w120597u푇 1205972119869ctls (uw)120597w120597w푇

]]]]


= Ψ1 (uw) +Ψ2 (uw) (27)

where

Ψ1 (uw) = [ 1205971205931 (uw)120597u푇 1205971205931 (uw)120597w푇 ]= [ Ψ11 (uw) Ψ12 (uw) ]


(28)

in which Ψ11(uw) = 1205971205931(uw)120597u푇 Ψ12(uw) = 1205971205931(uw)120597w푇 Ψ21(uw) = 1205971205932(uw)120597u푇 and Ψ22(uw) =1205971205932(uw)120597w푇 It follows from (26) that

Ψ11 (uw)

= [[[[[[2(g (uw) otimes 120597g (uw)120597u푇 )푇 sdot 120597vec (G (uw))120597u푇 + 2(120597g (uw)120597u푇 )푇G (uw) sdot 120597g (uw)120597u푇 + 2 (((g (uw))푇G (uw)) otimes I푝2퐷)( 120597120597u푇 vec((120597g (uw)120597u푇 )푇))2(g (uw) otimes 120597g (uw)120597w푇 )푇 sdot 120597vec (G (uw))120597u푇 + 2(120597g (uw)120597w푇 )푇G (uw) sdot 120597g (uw)120597u푇 + 2 (((g (uw))푇G (uw)) otimes I푝3)( 120597120597u푇 vec((120597g (uw)120597w푇 )푇))

]]]]]](29)

Ψ12 (uw)

= [[[[[[2(g (uw) otimes 120597g (uw)120597u푇 )푇 sdot 120597vec (G (uw))120597w푇 + 2(120597g (uw)120597u푇 )푇G (uw) sdot 120597g (uw)120597w푇 + 2 (((g (uw))푇G (uw)) otimes I푝2퐷)( 120597120597w푇 vec((120597g (uw)120597u푇 )푇))2(g (uw) otimes 120597g (uw)120597w푇 )푇 sdot 120597vec (G (uw))120597w푇 + 2(120597g (uw)120597w푇 )푇G (uw) sdot 120597g (uw)120597w푇 + 2 (((g (uw))푇G (uw)) otimes I푝3)( 120597120597w푇 vec((120597g (uw)120597w푇 )푇))

]]]]]] (30)

Ψ21 (uw) asymp [[[[[[(120597vec (G (uw))120597u푇 )푇 ((I푝1퐷+푝3 otimes g (uw)) sdot 120597g (uw)120597u푇 + g (uw) otimes 120597g (uw)120597u푇 )(120597vec (G (uw))120597w푇 )푇 ((I푝1퐷+푝3 otimes g (uw)) sdot 120597g (uw)120597u푇 + g (uw) otimes 120597g (uw)120597u푇 )

]]]]]](31)

Ψ22 (uw) asymp [[[[[[(120597vec (G (uw))120597u푇 )푇 ((I푝1퐷+푝3 otimes g (uw)) sdot 120597g (uw)120597w푇 + g (uw) otimes 120597g (uw)120597w푇 )(120597vec (G (uw))120597w푇 )푇 ((I푝1퐷+푝3 otimes g (uw)) sdot 120597g (uw)120597w푇 + g (uw) otimes 120597g (uw)120597w푇 )

]]]]]] (32)

It is worth pointing out that all the quadratic terms ofg(uw) are ignored in (31) and (32) The reason is that theseterms hardly affect the convergence rate and asymptoticperformance of the CTLS method

Based on the above discussion the Newton iteration isgiven by

[[u(푘+1)w(푘+1)

]] = [[u(푘)w(푘)

]]minus 120583푘 (Ψ(u(푘) w(푘)))minus1 120593(u(푘) w(푘))

(33)

where the subscript (119896) denotes the 119896th iteration and 120583 (0 lt120583 lt 1) is a suitable step size Some remarks on the Newtoniteration follow

Remark 2 The initial value for the iteration can be obtainedby the WLS or TLS methods both of which can provide anapproximate closed-form solution

Remark 3 120593(u(푘) w(푘))2 le 120585 can be used as the prescribedconvergence criterion

Remark 4 Note that in (29)ndash(32) there exist some matriceswhose expressions are not yet specified They include

Z1 = 120597g (uw)120597u푇 Z2 = 120597g (uw)120597w푇 Z3 = 120597120597u푇 vec((120597g (uw)120597u푇 )푇)Z4 = 120597120597w푇 vec((120597g (uw)120597u푇 )푇) Z5 = 120597120597u푇 vec((120597g (uw)120597w푇 )푇)Z6 = 120597120597w푇 vec((120597g (uw)120597w푇 )푇) Z7 = 120597vec (G (uw))120597u푇 Z8 = 120597vec (G (uw))120597w푇

(34)


The exact expressions for the eight matrices in (34) areprovided in Appendix A

Remark 5 Since the weighting matrix G(uw) = (Q(t zk))minus1 is updated at each iteration step the proposed CTLSsolution is able to yield much smaller estimation bias com-pared to the TLS and TWLS solutions as discussed inSection 7

5 Performance Analysis

In this section the analytical expression for the covariancematrix of the above CTLS estimator is derived Furthermorethe CTLS solution is proved theoretically to reach the CRBaccuracy before the thresholding effect starts to take place

51 Covariance Matrix of the CTLS Solution Assuming theconvergence results for the Newton iteration are denoted by

uctls and wctls it follows from the iteration termination criteriagiven in Remark 3 that

lim푘rarr+infin120593(u(푘) w(푘)) = 120593 (uctls wctls)

=[[[[[[[[[[

120597119869ctls (u wctls)120597u100381610038161003816100381610038161003816100381610038161003816u=uctls

120597119869ctls (uctlsw)120597w10038161003816100381610038161003816100381610038161003816100381610038161003816w=wctls

]]]]]]]]]]= O(푝2퐷+푝3)times1

(35)

The substitution of (25)-(26) into (35) leads to

O(푝2퐷+푝3)times1

=[[[[[[[[[

2( 120597g (u wctls)120597u푇100381610038161003816100381610038161003816100381610038161003816u=uctls)

푇

G (uctls wctls) g (uctls wctls) + ( 120597vec (G (u wctls))120597u푇100381610038161003816100381610038161003816100381610038161003816u=uctls)

푇 (g (uctls wctls) otimes g (uctls wctls))2( 120597g (uctlsw)120597w푇

10038161003816100381610038161003816100381610038161003816100381610038161003816w=wctls

)푇G (uctls wctls) g (uctls wctls) + ( 120597vec (G (uctlsw))120597w푇10038161003816100381610038161003816100381610038161003816100381610038161003816w=wctls

)푇 (g (uctls wctls) otimes g (uctls wctls))

]]]]]]]]] (36)

Performing a first-order Taylor series expansion ofg(uctls wctls) around the true values u and w produces

g (uctls wctls) asymp [[B (z0w) H1 (uw) sdot 120575uctls + B (z0w) H2 (uw) sdot 120575wctls minus C1 (t z0w) n minus C2 (t z0w)m

120575wctls minusm]]

= [B (z0w) H1 (uw) B (z0w) H2 (uw)O푝3times푝2퐷 I푝3

] sdot [120575uctls120575wctls

] minus [C1 (t z0w) C2 (t z0w)O푝3times푝1퐷 I푝3

] sdot [ nm]

(37)

where 120575uctls = uctls minus u and 120575wctls = wctls minus w are estimationerrors Besides H1(uw) = 120597h(uw)120597u푇 and H2(uw) =120597h(uw)120597w푇 whose expressions are given by

H1 (uw)= blkdiag [H1 (u1w) H1 (u2w) sdot sdot sdot H1 (u퐷w)]H2 (uw)= [(H2 (u1w))푇 (H2 (u2w))푇 sdot sdot sdot (H2 (u퐷w))푇]푇

(38)

where

H1 (u푑w) = 120597h (u푑w)120597u푇푑

= [ I푝2S1 (u푑w)]

isin R(푝2+푝4)times푝2

H2 (u푑w) = 120597h (u푑w)120597w푇 = [ minusJS2 (u푑w)]


(1 le 119889 le 119863)(39)

in which S1(u푑w) = 120597s(u푑w)120597u푇푑 isin R푝4times푝2 and S2(u푑w) =120597s(u푑w)120597w푇 isin R푝4times푝3 Substituting (37) into (36) and omitting the second- and

higher-order error terms yields



asymp [B (z0w) H1 (uw) B (z0w) H2 (uw)O푝3times푝2퐷 I푝3

]푇

sdot G0 (uw)sdot [B (z0w) H1 (uw) B (z0w) H2 (uw)

O푝3times푝2퐷 I푝3]

sdot [120575uctls120575wctls

]

minus [[B (z0w) H1 (uw) B (z0w) H2 (uw)

O푝3times푝2퐷 I푝3]]푇

sdot G0 (uw) sdot [[C1 (t z0w) C2 (t z0w)O푝3times푝1퐷 I푝3

]]sdot [ n

m]

(40)

where

G0 (uw) = G (uw)| n=O1199011119863times1m=O1199013times1

= (Q (t z0w))minus1

= [[[[C1 (t z0w) N (C1 (t z0w))푇 + C2 (t z0w)M (C2 (t z0w))푇 C2 (t z0w)M

M (C2 (t z0w))푇 M

]]]]

minus1

(41)

It can be readily deduced from (40) that

[120575uctls120575wctls

]

asymp ([B (z0w) H1 (uw) B (z0w) H2 (uw)O푝3times푝2퐷 I푝3

]푇


O푝3times푝2퐷 I푝3])minus1

times [B (z0w) H1 (uw) B (z0w) H2 (uw)O푝3times푝2퐷 I푝3

]푇

sdot G0 (uw) sdot [C1 (t z0w) C2 (t z0w)O푝3times푝1퐷 I푝3

] sdot [ nm]

(42)

Then the covariance matrix of estimated vector [ uctlswctls] is

given by

cok([uctlswctls

]) = 119864[[[120575uctls120575wctls


]푇]]= ([B (z0w) H1 (uw) B (z0w) H2 (uw)

O푝3times푝2퐷 I푝3]푇



(43)

52 Asymptotical Efficiency of the CTLSEstimator In order toprove that the aboveCTLS solution is asymptotically efficientit is necessary to obtain the correspondingCRB According to[14 16 24 28] we have

CRB([uw]) = [[[[

(F1 (uw))푇 Nminus1F1 (uw) (F1 (uw))푇 Nminus1F2 (uw)(F2 (uw))푇 Nminus1F1 (uw) (F2 (uw))푇 Nminus1F2 (uw) +Mminus1

]]]]

minus1

(44)

where F1(uw) = 120597f(uw)120597u푇 and F2(uw) = 120597f(uw)120597w푇Using the definition of f(uw) in (9) it is straightforward toshow that

F1 (uw)= blkdiag [F1 (u1w) F1 (u2w) sdot sdot sdot F1 (u퐷w)]


isin R푝1퐷times푝2퐷F2 (uw)= [(F2 (u1w))푇 (F2 (u2w))푇 sdot sdot sdot (F2 (u퐷w))푇]푇isin R푝1퐷times푝3

(45)

where F1(u푑w) = 120597f(u푑w)120597u푇푑 and F2(u푑w) = 120597f(u푑w)120597w푇

By comparing (43) and (44) we get the following propo-sition

Proposition 6 One has

cok([uctlswctls

]) = CRB([uw]) (46)

Proof First combining (41) and the matrix inversion formu-las leads to

G0 (uw)= [[[[

(C1 (t z0w))minus푇 Nminus1 (C1 (t z0w))minus1 minus (C1 (t z0w))minus푇 Nminus1 (C1 (t z0w))minus1 C2 (t z0w)minus (C2 (t z0w))푇 (C1 (t z0w))minus푇 Nminus1 (C1 (t z0w))minus1 Mminus1 + (C2 (t z0w))푇 (C1 (t z0w))minus푇 Nminus1 (C1 (t z0w))minus1 C2 (t z0w)

]]]] (47)

The proof of (47) is provided in Appendix BThe substitutionof (47) into (43) leads to

cok([uctlswctls

]) = [P1 P2P푇2 P3

]minus1 (48)

where

P1 = (H1 (uw))푇 (B (z0w))푇 (C1 (t z0w))minus푇sdot Nminus1 (C1 (t z0w))minus1 B (z0w) H1 (uw)

P2 = (H1 (uw))푇 (B (z0w))푇 (C1 (t z0w))minus푇sdot Nminus1 (C1 (t z0w))minus1sdot (B (z0w) H2 (uw) minus C2 (t z0w))

P3 = (B (z0w) H2 (uw) minus C2 (t z0w))푇sdot (C1 (t z0w))minus푇 Nminus1 (C1 (t z0w))minus1sdot (B (z0w) H2 (uw) minus C2 (t z0w)) +Mminus1

(49)

Next putting z푑0 = f(u푑w) into (4) producesa (f (u푑w) w) = B (f (u푑w) w) t푑

= B (f (u푑w) w) h (u푑w)(1 le 119889 le 119863)

(50)

Differentiating both sides of (50) with respect to u푑 andw wehaveA1 (z푑0w) F1 (u푑w)

= [B11 (z푑0w) t푑 B12 (z푑0w) t푑 sdot sdot sdot B1푝1 (z푑0w) t푑]sdot F1 (u푑w) + B (z푑0w)H1 (u푑w) 997904rArr

C1 (t푑 z푑0w) F1 (u푑w) = B (z푑0w)H1 (u푑w) 997904rArrF1 (u푑w) = (C1 (t푑 z푑0w))minus1 B (z푑0w)H1 (u푑w)

(51)

A1 (z푑0w) F2 (u푑w) + A2 (z푑0w)= [B11 (z푑0w) t푑 B12 (z푑0w) t푑 sdot sdot sdot B1푝1 (z푑0w) t푑]sdot F2 (u푑w)+ [B21 (z푑0w) t푑 B22 (z푑0w) t푑 sdot sdot sdot B2푝3 (z푑0w) t푑]+ B (z푑0w)H2 (u푑w) 997904rArr

C1 (t푑 z푑0w) F2 (u푑w) + C2 (t푑 z푑0w) = B (z푑0w)sdotH2 (u푑w) 997904rArr

F2 (u푑w) = (C1 (t푑 z푑0w))minus1sdot (B (z푑0w)H2 (u푑w) minus C2 (t푑 z푑0w))

(52)

whereC1 (t푑 z푑0w) = A1 (z푑0w)

minus [B11 (z푑0w) t푑 B12 (z푑0w) t푑 sdot sdot sdot B1푝1 (z푑0w) t푑]isin R푝1times푝1

C2 (t푑 z푑0w) = A2 (z푑0w)minus [B21 (z푑0w) t푑 B22 (z푑0w) t푑 sdot sdot sdot B2푝3 (z푑0w) t푑]isin R푝1times푝3

(53)


in which

A1 (z푑0w) = 120597a (z푑0w)120597z푇푑0

isin R푝1times푝1 A2 (z푑0w) = 120597a (z푑0w)120597w푇 isin R푝1times푝3

B1푗 (z푑0w) = 120597B (z푑0w)120597 ⟨z푑0⟩푗 isin R푝1times(푝2+푝4)

(1 le 119895 le 1199011)B2푗 (z푑0w) = 120597B (z푑0w)120597 ⟨w⟩푗 isin R푝1times(푝2+푝4)

(1 le 119895 le 1199013) (54)

From (11) (13) (53) and (54) it can be verified that

C1 (t z0w) = blkdiag [C1 (t1 z10w) C1 (t2 z20w) sdot sdot sdot C1 (t퐷 z퐷0w)] C2 (t z0w) = [(C2 (t1 z10w))푇 (C2 (t2 z20w))푇 sdot sdot sdot (C2 (t퐷 z퐷0w))푇]푇 (55)

Combining the second equality in (7) the first equality in(38) the first equality in (45) and the first equality in (55)and (51) yields

F1 (uw) = (C1 (t z0w))minus1 B (z0w) H1 (uw) (56)

Putting the second equality in (7) the second equality in (38)the second equality in (45) and the second equality in (55)and (52) produces

F2 (uw) = (C1 (t z0w))minus1sdot (B (z0w) H2 (uw) minus C2 (t z0w)) (57)

The substitution of (56) and (57) into (49) leads to

P1 = (F1 (uw))푇 Nminus1F1 (uw)P2 = (F1 (uw))푇 Nminus1F2 (uw)P3 = (F2 (uw))푇 Nminus1F2 (uw) +Mminus1

(58)

which combined with (44) and (48) completes the proof

Remark 7 It can be concluded from Proposition 6 that theproposed CTLS solution is able to achieve the CRB accu-racy at moderate noise level before the thresholding effectdue to nonlinear nature of the estimation problem occursMoreover it is worth stressing that different from the existingtheoretical analysis in the literature the proof describedabove is independent on the specific measurement type usedIn other words it can be considered as a more generalanalysis framework which is suitable formany different loca-tion measurements Furthermore the experiment results inSection 7 show that the proposed method can tolerate highernoise level compared to the other location methods

Remark 8 It is known that the performance of TLS estimatorcan be dependent on how the origin of the coordinatesystem is selected as shown in [37] It is obvious that suchdependence also exists for the proposed CTLS method

Indeed it is a valuable research topic and is worthy of furtherinvestigation However it is not a trivial task because themathematical analysis is rather elaborate Hence we can con-sider it as an open problem which will be focused on in ourfuture study

6 Two Localization Examples

In this section two localization scenarios are discussed inorder to show how to exploit the proposed CTLS methodto locate multiple disjoint sources and improve the sensorpositions simultaneously

61 Sources Localization Using TDOA and GROA Measure-ments Assume that there are 119863 static and disjoint sourcesto be localized using a wireless location system that contains119870 stationary sensors The unknown source positions aredenoted by u푑 = [119909푡푑 119910푡푑 119911푡푑]푇 (1 le 119889 le 119863) The exactsensor position is represented byw푘 = [119909표푘 119910표푘 119911표푘]푇 (1 le119896 le 119870) and then the system parameter is given byw = [w푇1 w푇2 sdot sdot sdot w푇퐾]푇 As previously assumed the systemparameter available for processing denoted by k has randomerrors The TDOA and GROA measurements with respectto the reference sensor say sensor 1 are exacted from thereceived signals Note that the TDOA and GROA mea-surements are equivalent to the range difference and rangeratio measurements respectively As a consequence thecorresponding observation equations are given by120588푑푘 = 1003817100381710038171003817u푑 minus w푘

10038171003817100381710038172 minus 1003817100381710038171003817u푑 minus w110038171003817100381710038172

119903푑푘 = 1003817100381710038171003817u푑 minus w푘100381710038171003817100381721003817100381710038171003817u푑 minus w110038171003817100381710038172

(2 le 119896 le 1198701 le 119889 le 119863) (59)

Let us define the following vectors

120588푑 = [120588푑2 120588푑3 sdot sdot sdot 120588푑퐾]푇 r푑 = [119903푑2 119903푑3 sdot sdot sdot 119903푑퐾]푇 (60)


Then the noiseless measurement vector related to the 119889thsource is given by

z푑0 = [120588푇푑 r푇푑]푇 = f (u푑w) isin R2(퐾minus1)times1

(1 le 119889 le 119863) (61)

The collection of all themeasurements forms the 2(119870minus1)119863times1vector as follows

z0 = [z푇10 z푇20 sdot sdot sdot z푇퐷0]푇= [(f (u1w))푇 (f (u2w))푇 sdot sdot sdot (f (u퐷w))푇]푇= f (uw) isin R2(퐾minus1)퐷times1

(62)

In order to employ the proposed CTLSmethod for sourcelocalization we must transform the nonlinear equationsin (59) into the pseudo-linear ones by introducing someauxiliary variables First it follows from the first equation in(59) that

120588푑푘 = 1003817100381710038171003817u푑 minus w푘10038171003817100381710038172 minus 1003817100381710038171003817u푑 minus w1

10038171003817100381710038172 997904rArr(120588푑푘 + 1003817100381710038171003817u푑 minus w1

10038171003817100381710038172)2 = 1003817100381710038171003817u푑 minus w1 + w1 minus w푘100381710038171003817100381722 997904rArr

2 (w1 minus w푘)푇 (u푑 minus w1) minus 2120588푑푘 sdot 1003817100381710038171003817u푑 minus w110038171003817100381710038172

= 1205882푑푘 minus 1003817100381710038171003817w1 minus w푘100381710038171003817100381722 997904rArr

(b1푘 (z푑0w))푇 sdot [ u푑 minus w11003817100381710038171003817u푑 minus w110038171003817100381710038172] = 1198861푘 (z푑0w)(2 le 119896 le 119870 1 le 119889 le 119863)

(63)

where

b1푘 (z푑0w) = [ 2 (w1 minus w푘)푇 minus2120588푑푘 ]푇1198861푘 (z푑0w) = 1205882푑푘 minus 1003817100381710038171003817w1 minus w푘

100381710038171003817100381722 (64)

At the same time we also obtain from (59) that

119903푑푘 = 1003817100381710038171003817u푑 minus w푘100381710038171003817100381721003817100381710038171003817u푑 minus w110038171003817100381710038172 997904rArr1003817100381710038171003817u푑 minus w푘

10038171003817100381710038172 = 119903푑푘 sdot 1003817100381710038171003817u푑 minus w110038171003817100381710038172 997904rArr

(119903푑푘 minus 1) sdot 1003817100381710038171003817u푑 minus w110038171003817100381710038172 = 1003817100381710038171003817u푑 minus w푘

10038171003817100381710038172 minus 1003817100381710038171003817u푑 minus w110038171003817100381710038172= 120588푑푘 997904rArr


(65)

where

b2푘 (z푑0w) = [ O1times3 119903푑푘 minus 1 ]푇1198862푘 (z푑0w) = 120588푑푘 (66)

Combining (63)ndash(66) yields the following pseudo-linearvector equation


where

a (z푑0w) = [(a1 (z푑0w))푇 (a2 (z푑0w))푇]푇B (z푑0w) = [(B1 (z푑0w))푇 (B2 (z푑0w))푇]푇

t푑 = h (u푑w) = [ u푑 minus w11003817100381710038171003817u푑 minus w110038171003817100381710038172] = [u푑 minus Jw

s (u푑w)](68)

in which

B푗 (z푑0w) =[[[[[[[[

b푇푗2 (z푑0w)b푇푗3 (z푑0w)b푇푗퐾 (z푑0w)

]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗2 (z푑0w)119886푗3 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]](1 le 119895 le 2)

s (u푑w) = 1003817100381710038171003817u푑 minus w110038171003817100381710038172

J = [I3 O3times3(퐾minus1)]

(69)

Collecting all the119863 vector equations in (67) gives

a (z0w) = B (z0w) t = B (z0w) h (uw) (70)

where

a (z0w)= [(a (z10w))푇 (a (z20w))푇 sdot sdot sdot (a (z퐷0w))푇]푇

B (z0w)= blkdiag [B (z10w) B (z20w) sdot sdot sdot B (z퐷0w)]

t = h (uw) = [t푇1 t푇2 sdot sdot sdot t푇퐷]푇= [(h (u1w))푇 (h (u2w))푇 sdot sdot sdot (h (u퐷w))푇]푇

z0 = [z푇10 z푇20 sdot sdot sdot z푇퐷0]푇 u = [u푇1 u푇2 sdot sdot sdot u푇퐷]푇

(71)


According to the discussions in the previoussections and in Appendix A we also need to derive theexpressions for C1(t푑 z푑0w) C2(t푑 z푑0w) H1(u푑w)H2(u푑w) 120597vec((S1(u푑w))푇)120597u푇푑 120597vec((S2(u푑w))푇)120597u푇푑 120597vec((S1(u푑w))푇)120597w푇 and 120597vec((S2(u푑w))푇)120597w푇 Thedetailed derivations of them are provided in Appendix C Atthis point the CTLS localization method using TDOA andGROA measurements can be summarized as follows

Step 1 Compute B(z k) and a(z k) according to (63)ndash(71)and determine the initial values by WLS or TLS methods

Step 2 Compute C1(t z k) and C2(t z k) using (55) as wellas the expressions for C1(t푑 z푑0w) and C2(t푑 z푑0w) andcalculate G(uw) according to (18) and (24)

Step 3 Compute H1(uw) and H2(uw) from (38) (39)(A3) and (A4) as well as the expressions for H1(u푑w) andH2(u푑w) and calculate Z1 and Z2 using (A1) and (A2)

Step 4 Compute Z3 Z4 Z5 and Z6 from (A5)ndash(A13) as wellas the expressions for 120597vec((S1(u푑w))푇)120597u푇푑 120597vec((S2(u푑w))푇)120597u푇푑 120597vec((S1(u푑w))푇)120597w푇 and 120597vec((S2(u푑w))푇)120597w푇Step 5 Compute Z7 and Z8 according to (A14)ndash(A21)

Step 6 Compute gradient 120593(uw) from (25)-(26) and com-pute Hessian matrixΨ(uw) according to (27)ndash(32)Step 7 If the predefined convergence criterion is satisfiedthen terminate otherwise update the unknowns using (33)and continue with Step 2

62 Sources Localization Using TOA and FOAMeasurementsIt is assumed that there are 119863 moving and disjoint sourcesto be located by a wireless location system that is composedof 119870 moving sensors The position and velocity of the119889th source are denoted by u푑푝 = [119909푡푑 119910푡푑 119911푡푑]푇 andu푑V = [푡푑 119910푡푑 푡푑]푇 respectively The location parameterof source 119889 is defined by u푑 = [u푇푑푝 u푇푑V]푇 The true positionand velocity of the 119896th sensor are represented by w푘푝 =[119909표푘 119910표푘 119911표푘]푇 and w푘V = [표푘 119910표푘 표푘]푇 respectivelyDefine w푘 = [w푇푘푝 w푇푘V]푇 and then the system parameter is

given by w = [w푇1 w푇2 sdot sdot sdot w푇퐾]푇 Note that the vector w isnot known exactly and only the noisy version of it denotedby k is available in practice Additionally the TOA andFOA measurements are obtained from the observed signalsSince the TOA and FOA measurements are equivalent to therange and range rate measurements respectively the relevantobservation equations can be expressed by

120583푑푘 = 10038171003817100381710038171003817u푑푝 minus w푘푝100381710038171003817100381710038172

120583푑푘 = (u푑푝 minus w푘푝)푇 (u푑V minus w푘V)10038171003817100381710038171003817u푑푝 minus w푘푝100381710038171003817100381710038172

(1 le 119896 le 119870) (72)

Define the following vectors

120583푑 = [120583푑1 120583푑2 sdot sdot sdot 120583푑퐾]푇 푑 = [푑1 120583푑2 sdot sdot sdot 120583푑퐾]푇 (73)

Then the noiseless measurement vector associated with the119889th source is given by

z푑0 = [120583푇푑 푇푑]푇 = f (u푑w) isin R2퐾times1 (1 le 119889 le 119863) (74)

Gathering all the measurements in a 2119870119863 times 1 vector leads toz0 = [z푇10 z푇20 sdot sdot sdot z푇퐷0]푇= [(f (u1w))푇 (f (u2w))푇 sdot sdot sdot (f (u퐷w))푇]푇= f (uw) isin R2퐾퐷times1

(75)

To make use of the presented CTLS localization methodwe must convert the nonlinear equations in (72) into thepseudo-linear ones by introducing some instrumental vari-ables From the first equation in (72) we have

120583푑푘 = 10038171003817100381710038171003817u푑푝 minus w푘푝100381710038171003817100381710038172 997904rArr

2w푇푘푝u푑푝 minus 10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722 = 10038171003817100381710038171003817w푘푝1003817100381710038171003817100381722 minus 1205832푑푘 997904rArr

(b1푘 (z푑0w))푇 sdot[[[[[[[

u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198861푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(76)

where

b1푘 (z푑0w) = [ 2w푇푘푝 O1times3 minus1 0 ]푇 1198861푘 (z푑0w) = 10038171003817100381710038171003817w푘푝1003817100381710038171003817100381722 minus 1205832푑푘

(77)

Taking the time derivation of the second equation in (76)leads to

2w푇푘푝u푑푝 minus 10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722 = 10038171003817100381710038171003817w푘푝1003817100381710038171003817100381722 minus 1205832푑푘 997904rArrw푇푘Vu푑푝 + w푇푘푝u푑V minus u푇푑Vu푑푝 = w푇푘Vw푘푝 minus 120583푑푘 120583푑푘 997904rArr


u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198862푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(78)


where

b2푘 (z푑0w) = [ w푇푘V w푇푘푝 0 minus1 ]푇 1198862푘 (z푑0w) = w푇푘Vw푘푝 minus 120583푑푘 120583푑푘 (79)

Putting (76)ndash(79) together gives the following pseudo-linearvector equation


where


t푑 = h (u푑w) = [[[[u푑10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]= [ u푑 minus Jw

s (u푑w)](81)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗1 (z푑0w)119886푗2 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]]

(1 le 119895 le 2)s (u푑w) = [[

10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722u푇푑Vu푑푝

]] J = O6times6퐾

(82)

It can be readily seen from (81) and (82) that neitherh(u푑w) nor s(u푑w) is dependent on w for the localizationscenario under discussion This leads to a reduction of thecomputation load Putting (80) together for 119889 = 1 2 119863gives

a (z0w) = B (z0w) t = B (z0w) h (uw) (83)

where


Table 1 Nominal positions of sensors

Sensor number 119896 119909표푘 (m) 119910표푘 (m) 119911표푘 (m)(1) 1800 minus2000 1200(2) minus1400 1800 1600(3) 1700 minus1400 minus1500(4) minus1100 1300 minus1800(5) 1800 1500 2100(6) minus1900 minus1200 minus1700B (z0w)= blkdiag [B (z10w) B (z20w) sdot sdot sdot B (z퐷0w)] t = h (uw) = [t푇1 t푇2 sdot sdot sdot t푇퐷]푇= [(h (u1w))푇 (h (u2w))푇 sdot sdot sdot (h (u퐷w))푇]푇 z0 = [z푇10 z푇20 sdot sdot sdot z푇퐷0]푇 u = [u푇1 u푇2 sdot sdot sdot u푇퐷]푇

(84)

Based on the discussions in the previous sectionsand in Appendix A we also need to derive theexpressions for C1(t푑 z푑0w) C2(t푑 z푑0w) H1(u푑w)H2(u푑w) 120597vec((S1(u푑w))푇)120597u푇푑 120597vec((S2(u푑w))푇)120597u푇푑 120597vec((S1(u푑w))푇)120597w푇 and 120597vec((S2(u푑w))푇)120597w푇 Thedetailed derivations of them are shown in Appendix D Onthe other hand it is obvious that the algorithm describedin Section 61 is applicable here and we thus omit it due tolimited space

7 Simulations Results

In this section some computer simulations are reported toillustrate the behavior of the presented method The root-mean-square-error (RMSE) and norm of bias are chosen asperformance metrics All the simulation results are averagedover 5000 independent noise realizations The proposedsolution is implemented using the procedure described inSection 6 The initial value of this iterative algorithm is givenby the WLS method the result of which is equal to the first-step estimate of the TWLS method

71 Numerical Results for TDOAsGROAs Source LocalizationIn this subsection the simulations are performed for sourcelocalization using TDOA and GROA measurements Theestimation accuracy of the proposed CTLS algorithm iscompared to that of the TLS algorithm and the TWLSalgorithm as well as the corresponding CRB given by (44)Additionally in order to show the cooperation gain resultingfrom joint localization formultiple sources theCRBobtainedfrom (44) is also compared to the CRB for the case in whichthe sources are located independently

The localization scenario contains 6 sensors andtheir nominal positions are given in Table 1 The


TLS solutionTWLS solutionProposed CTLS solutionCRB (locating the sources separately)CRB (given by (44))

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0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)

Figure 1 RMSE of the estimated position for the first source versus1205901

noisy sensor locations are created by adding to thetrue values zero-mean white Gaussian noise withcovariance matrix M = 1205902푤I3퐾 There are three disjointsources to be located and their true positions are u1 =[6000 6000 3000]푇 (m) u2 = [6500 6500 3500]푇 (m)and u3 = [7000 7000 4000]푇 (m) The TDOAs andGROAs for a given source are generated by adding thezero-mean Gaussian noise to the true values The covariancematrix is N푑 = blkdiag [1205902TDOAR 1205902GROAR] where R is a(119870 minus 1) times (119870 minus 1) matrix with diagonal elements equal to 1and all other elements 05 The measurements from differentsources are independent of each other Consequently thecovariance matrix N is block diagonal

In the first experiment we fix 120590푤 = 5 and set 120590TDOA =0151205901119888 120590GROA = 000151205901 where 1205901 varies from 1 to 20 and119888 is the signal propagation speed Figures 1ndash3 respectivelydisplay the RMSE of position estimates for the three sourcesversus 1205901 Figure 4 plots the RMSE of the estimated receivingposition as a function of 1205901 In the second experiment wefix 120590TDOA = 1119888 120590GROA = 001 and set 120590푤 = 061205902 where 1205902ranges from 1 to 20 Figures 5ndash7 respectively show the RMSEof location estimates for the three sources versus 1205902 Figure 8illustrates the RMSE of the estimated receiving position as afunction of 1205902

It can be seen from Figures 1ndash8 that the proposedCTLS method can reach the CRB given by (44) undermoderate noise level which demonstrates the validity of theperformance analysis in Section 5 Moreover the proposedmethod achieves noticeably better accuracy than the TLSmethod and it has a higher noise threshold than the TWLSmethod By comparing the two kinds of CRB we can findthat the performance improvement due to joint localization


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20

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60

80

100

120

140

160

180

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)Figure 2 RMSE of the estimated position for the second sourceversus 1205901


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

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100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)

Figure 3 RMSE of the estimated position for the third source versus1205901is remarkable Moreover it can be observed from Figures5ndash7 that the cooperation gain increases with increasing 1205902The reason is that as 1205902 increases the correlation betweenthe measurements of distinct sources becomes larger andconsequently the effect of cooperative processing can bemoreconsiderable On the other hand we can also find fromFigures 4 and 8 that compared to the prior knowledge of the

Mathematical Problems in Engineering 15Po

sitio

n RM

SE o

f the

sens

ors (

m)

TLS solutionTWLS solutionProposed CTLS solutionPrior RMSECRB (given by (44))

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

20

205

21

215

Figure 4 RMSE of the sensor position estimate versus 1205901


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)

Figure 5 RMSE of the estimated position for the first source as afunction of 1205902sensor positions the proposed CTLS method can provide anestimate with a smaller RMSE

In the following experiments we compare the norm ofsource position bias of the proposed CTLS method with thatof the TWLS method The simulation parameters are set asthe same as previously described except that the standarddeviations of noises are changed


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)Figure 6 RMSE of the estimated position for the second source asa function of 1205902


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)

Figure 7 RMSE of the estimated position for the third source as afunction of 1205902

First we let 120590푤 = 20 and set 120590TDOA = 031205901119888 120590GROA =00031205901 where 1205901 varies from 1 to 20 Figure 9 depicts thenorm of source position bias for the three sources as afunction of 1205901 Next we choose 120590TDOA = 3119888 120590GROA = 003and set120590푤 = 21205902 where1205902 ranges from 1 to 20 Figure 10 plotsthe norm of source position bias for the three sources versus1205902



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)

Figure 8 RMSE of the sensor position estimate as a function of 1205902

TWLS solution (the first source)TWLS solution (the second source)TWLS solution (the third source)Proposed CTLS solution (the first source)Proposed CTLS solution (the second source)Proposed CTLS solution (the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)

Figure 9 Norm of source position bias as a function of 1205901Figures 9 and 10 show that the estimation bias of the

proposed CTLS method is much smaller than that of theTWLS method especially when the noise is large Thisobservation is not unexpected because as studied in [38] theTWLS method has large estimation bias at high noise levelAdditionally the proposed CTLS method can yield relativelysmall deviation as expected The reason lies in that it canremove the bias by updating the weighting matrix in the


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)

Figure 10 Norm of source position bias as a function of 1205902

iterative process as stated in Remark 5 On the other handin order to prevent the curves in Figures 9 and 10 from beingmessy the estimation bias of the TLS method is not includedin Figures 9 and 10 Indeed the bias of this method is foundto be considerably larger than that of the other two methodsdue to the fact that the TLS estimator is biased as studied in[1]

72 Numerical Results for TOAsFOAs Source LocalizationIn this subsection the simulations are carried out for thelocalization scenario using TOA and FOAmeasurementsWecompare the performance of the proposed CTLS algorithmwith the TLS algorithm and the TS algorithm as well asthe corresponding CRB computed by (44) Besides for thepurpose of showing the advantage of cooperative localizationthe CRB for the case of single-source location is displayedagain On the other hand it is noteworthy that the TS algo-rithm requires initial solution guess However good initialestimate is not easily available for this algorithm because itdoes not provide the pseudo-linear vector equation For acomprehensive comparison the TS algorithm is initialized intwo ways One chooses random value as initial guess and theother takes the true value as initial solution

In the following simulation an array of 6 sensorsis used to locate the disjoint sources and the nominalpositions and velocities of sensors are listed in Table 2The sensor location and velocity errors follow zero-mean Gaussian distribution with covariance matrixM = I퐾 otimes blkdiag [1205902푤119901I3 1205902푤VI3] Three moving sourcesneed to be located Their true positions are u1푝 =[6000 6000 3000]푇 (m) u2푝 = [6500 6500 3500]푇 (m)


Table 2 Nominal positions and velocities of sensors

Sensor number 119896 119909표푘 (m) 119910표푘 (m) 119911표푘 (m) 표푘 (ms) 119910표푘 (ms) 표푘 (ms)(1) 2000 minus2000 1500 20 minus30 10(2) minus1400 1800 1900 minus10 minus10 20(3) 1400 minus1500 minus1600 20 30 minus10(4) minus1300 1400 minus1300 10 20 10(5) 1600 1800 2000 minus20 minus10 minus30(6) minus1700 minus1400 minus1600 minus10 20 20

TLS solutionTS solution with random initializationTS solution without initial errorsProposed CTLS solutionCRB (locating the sources separately)CRB (given by (44))

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40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)


and u3푝 = [7000 7000 4000]푇 (m) Their exact velocitiesare u1V = [10 minus20 20]푇 (ms) u2V = [20 20 10]푇 (ms)and u3V = [30 10 minus10]푇 (ms) The TOAFOA meas-urement errors for a given source are zero-meanGaussian distributed with covariance matrix N푑 =blkdiag [1205902TOAI퐾 1205902FOAI퐾] The measurements from differentsources are assumed to be uncorrelated with each otherwhich leads to a block-diagonal structure of the covariancematrix N

In the first experiment we fix 120590푤푝 = 20 120590푤V = 05 andset 120590TOA = 1205901119888 120590FOA = 00111989101205901119888 where 1205901 is changedfrom 1 to 20 and 1198910 is the signal carrier frequency Figures11ndash13 respectively plot the RMSE of position estimates forthe three sources versus 1205901 Figures 14ndash16 respectively plotthe RMSE of velocity estimates for the three sources versus1205901 Figures 17 and 18 plot the RMSE of position and velocityestimates for the sensor as a function of 1205901 respectively

In the second experiment we fix 120590TOA = 3119888 120590FOA =0011198910119888 and set 120590푤푝 = 151205902 120590푤V = 0051205902 where 1205902 varies


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)

Figure 12 RMSE of the estimated velocity for the first source versus1205901

from 1 to 20 Figures 19ndash21 respectively illustrate the RMSEof position estimates for the three sources versus 1205902 Figures22ndash24 respectively show the RMSE of velocity estimates forthe three sources versus 1205902 Figures 25 and 26 plot the RMSEof position and velocity estimates for the sensor as a functionof 1205902 respectively

It can be observed from Figures 11ndash26 that the pro-posed CTLS solution achieves the CRB accuracy given by(44) under moderate noise level which can corroboratethe theoretical comparison between the performance of theproposed estimator and the CRB The proposed method stilloutperforms the TLS method for this localization scenarioThe advantage of cooperation localization is also noticeableby comparing the two kinds of CRB Additionally if theTS algorithm is initialized with the true value it yields anearly similar performance to the proposed CTLS algorithmHowever when the initial value of the TS algorithm ischosen randomly its performance deviates from the CRBearlier compared to the CTLS algorithm Indeed it is hard



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50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)

Figure 13 RMSE of the estimated position for the second sourceversus 1205901


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)

Figure 14 RMSE of the estimated velocity for the second sourceversus 1205901to get a good initial estimate for the TS algorithm becausethis algorithm does not yield a pseudo-linear measurementequation On the other hand it can also be seen from Figures17 18 25 and 26 that the proposed method can improve theestimation accuracy for the sensor locations in comparison toits prior position information


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60

70

80

90

100

110

120

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ion

RMSE

of t

he th

ird so

urce

(m)

Figure 15 RMSEof the estimated velocity for the third source versus1205901


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)

Figure 16 RMSE of the estimated velocity for the third sourceversus 1205901

In the following experiments we compare the norm ofsource position bias of the proposed CTLS solution withthat of the TS algorithm which is randomly initialized Thesimulation parameters are assumed the same as those statedabove except that we change the standard deviations ofnoises


TLS solutionTS solution with random initializationTS solution without initial errorsProposed CTLS solutionPrior RMSECRB (given by (44))

2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)

Figure 18 RMSE of the sensor velocity estimate versus 1205901We fix 120590푤푝 = 20 120590푤V = 1 and set 120590TOA = 2120590119888 120590FOA =0021198910120590119888 where 120590 varies from 1 to 20 Figures 27 and 28

depict the norm of source position and velocity bias for thethree sources versus 120590 respectively

Figures 27 and 28 demonstrate that the proposed CTLSalgorithm can yield very small estimation bias Moreoverthe bias of the CTLS solution is very close to that of the TS


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)

Figure 20 RMSE of the estimated velocity for the first source versus1205902algorithm which is initialized with the true value But theresult of the latter is not displayed in Figures 27 and 28because it wouldmake the curves rather confusing Addition-ally as shown in Figures 27 and 28 if the TS algorithm is ran-domly initialized its estimation bias increases suddenlywhenthe noise level exceeds a certain threshold This observationis consistent with the conclusion stated above



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)

Figure 22 RMSE of the estimated velocity for the second sourceversus 12059028 Conclusions

In this paper we present an efficient CTLS method thatcan locate multiple disjoint sources and refine the erroneoussensor positions simultaneously Unlike the conventionallocalization methods an important feature of the proposedmethod is that it establishes a general framework that


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)

Figure 24 RMSE of the estimated velocity for the third sourceversus 1205902is suitable for many different location measurements Amodified CTLS optimization problem is formulated aftersome algebraic manipulations and the corresponding New-ton iterative algorithm is also derived to yield the numericalsolution Besides by exploiting the first-order perturbationanalysis the exact expression for the covariance matrix of theproposed CTLS estimator is derived under the Gaussian



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)

Figure 26 RMSE of the sensor velocity estimate versus 1205902

assumption The estimation accuracy of the CTLS methodis proved to achieve the CRB before the thresholding effectstarts to take place Additionally two examples are given toexplain how to utilize the proposed CTLS method for sourcelocalization One uses the TDOAsGROAs measurementsand the other is based on the TOAsFOAs parameters Sim-ulation results verify the good performance of the proposed

TS solution with random initialization



Proposed CTLS solution (the first source)Proposed CTLS solution (the second source)Proposed CTLS solution (the third source)

(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)

Figure 28 Norm of source velocity bias as a function of 120590method and also corroborate the performance analysis in thispaper Finally it is worth emphasizing that the theoretical


development in this paper is not limited to some specificmeasurements and it can be applied to many localizationscenarios as long as the measurement equation can betransformed into the pseudo-linear model

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors acknowledge support fromNational Natural Sci-ence Foundation of China (Grant no 61201381 no 61401513and no 61772548) China Postdoctoral Science Foundation(Grant no 2016M592989) the Self-Topic Foundation ofInformation Engineering University (Grant no 2016600701)and the Outstanding Youth Foundation of Information Engi-neering University (Grant no 2016603201)

Supplementary Materials

Supplementary materials contain four appendices whichare called Appendix A Appendix B Appendix C andAppendix D respectively (Supplementary Materials)

References

[1] K Dogancay ldquoBearings-only target localization using total leastsquaresrdquo Signal Processing vol 85 no 9 pp 1695ndash1710 2005

[2] X Lu and K C Ho ldquoTaylor-series technique for source local-ization using AoAs in the presence of sensor location errorsrdquoin Proceedings of the 4th IEEE Sensor Array and MultichannelSignal ProcessingWorkshop Proceedings SAM2006 pp 190ndash194usa July 2006

[3] D Wang L Zhang and Y Wu ldquoConstrained total leastsquares algorithm for passive location based on bearing-onlymeasurementsrdquo Science China Information Sciences vol 50 no4 pp 576ndash586 2007

[4] K W Cheung H C So W-K Ma and Y T Chan ldquoLeastsquares algorithms for time-of-arrival-based mobile locationrdquoIEEE Transactions on Signal Processing vol 52 no 4 pp 1121ndash1128 2004

[5] Z Ma and K C Ho ldquoTOA localization in the presence ofrandom sensor position errorsrdquo in Proceedings of the 36thIEEE International Conference on Acoustics Speech and SignalProcessing ICASSP 2011 pp 2468ndash2471 Czech Republic May2011

[6] Y Zhou J Li and L Lamont ldquoMultilateration localization inthe presence of anchor location uncertaintiesrdquo in Proceedings ofthe IEEEGlobal Communications Conference (GLOBECOM rsquo12)pp 309ndash314 December 2012

[7] M Sun Z Ma and K C Ho ldquoJoint source localization and sen-sor position refinement for sensor networksrdquo in Proceedings ofthe 2013 38th IEEE International Conference on AcousticsSpeech and Signal Processing ICASSP 2013 pp 4026ndash4030Canada May 2013

[8] Y T Chan and K C Ho ldquoA simple and efficient estimator forhyperbolic locationrdquo IEEE Transactions on Signal Processingvol 42 no 8 pp 1905ndash1915 1994

[9] Y Huang J Benesty G W Elko and R M Mersereau ldquoReal-time passive source localization a practical linear-correctionleast-squares approachrdquo IEEE Transactions on Audio Speechand Language Processing vol 9 no 8 pp 943ndash956 2001

[10] H C So and S P Hui ldquoConstrained Location Algorithm UsingTDOAMeasurementsrdquo IEICE Transactions on Fundamentals ofElectronics Communications and Computer Sciences vol E86-A no 12 pp 3291ndash3293 2003

[11] Z Huang and J Lu ldquoTotal least squares and equilibration algo-rithm for range difference locationrdquo IEEE Electronics Lettersvol 40 no 5 pp 323ndash325 2004

[12] L Kovavisaruch and K C Ho ldquoModified Taylor-series Methodfor Source and Receiver Localization Using TDOA Measure-ments with Erroneous Receiver Positionsrdquo in Proceedings of theIEEE International Symposium on Circuits and Systems 2005ISCAS 2005 pp 2295ndash2298 jpn May 2005

[13] Y Zhou and L Lamont ldquoConstrained linear least squaresapproach for tdoa localization A global optimum solutionrdquoin Proceedings of the 2008 IEEE International Conference onAcoustics Speech and Signal Processing ICASSP pp 2577ndash2580USA April 2008

[14] L Yang and K C Ho ldquoAn approximately efficient TDOA local-ization algorithm in closed-form for locating multiple disjointsources with erroneous sensor positionsrdquo IEEE Transactions onSignal Processing vol 57 no 12 pp 4598ndash4615 2009

[15] K Yang J An X Bu and G Sun ldquoConstrained total least-squares location algorithm using time-difference-of-arrivalmeasurementsrdquo IEEETransactions onVehicular Technology vol59 no 3 pp 1558ndash1562 2010

[16] M Sun L Yang and D K C Ho ldquoEfficient joint sourceand sensor localization in closed-formrdquo IEEE Signal ProcessingLetters vol 19 no 7 pp 399ndash402 2012

[17] S Chen H He and H Yu ldquoConstrained total least-squares forsource location using TDOA measurements in the presence ofsensor position errorsrdquo Aeronautica et Astronautica Sinica vol34 no 5 pp 1165ndash1173 2013

[18] J Mason ldquoAlgebraic two-satellite TOAFOA position solutionon an ellipsoidal earthrdquo IEEE Transactions on Aerospace andElectronic Systems vol 40 no 3 pp 1087ndash1092 2004

[19] K C Ho andW Xu ldquoAn accurate algebraic solution for movingsource location using TDOA and FDOA measurementsrdquo IEEETransactions on Signal Processing vol 52 no 9 pp 2453ndash24632004

[20] X N Lu and K C Ho ldquoTaylor-series technique for movingsource localization in the presence of sensor location errorsrdquoin Proceedings of the 2006 IEEE International Symposium onCircuits and Systems pp 1075ndash1078 Island of Kos Greece 2006

[21] K C Ho X Lu and L Kovavisaruch ldquoSource localization usingTDOA and FDOA measurements in the presence of receiverlocation errors analysis and solutionrdquo IEEE Transactions onSignal Processing vol 55 no 2 pp 684ndash696 2007

[22] S Xiaoyan L Jiandong H Pengyu and P Jiyong ldquoTotal least-squares solution of active target localization using TDOA andFDOA measurements in WSNrdquo in Proceedings of the 22ndInternational Conference on Advanced Information Networkingand Applications WorkshopsSymposia AINA 2008 pp 995ndash999 Japan March 2008

[23] H Wu W-M Su and H Gu ldquoA novel Taylor series method forsource and receiver localization using TDOA and FDOA mea-surements with uncertain receiver positionsrdquo in Proceedings ofthe 6th International Conference on Radar RADAR 2011 pp1037ndash1040 China October 2011


[24] M Sun and K C Ho ldquoAn asymptotically efficient estimator forTDOA and FDOA positioning of multiple disjoint sources inthe presence of sensor location uncertaintiesrdquo IEEE Transac-tions on Signal Processing vol 59 no 7 pp 3434ndash3440 2011

[25] H G Yu G M Huang J Gao and B Liu ldquoAn efficientconstrainedweighted least squares algorithm formoving sourcelocation using TDOA and FDOA measurementsrdquo IEEE Trans-actions on Wireless Communications vol 11 no 1 pp 44ndash472012

[26] H Yu G Huang and J Gao ldquoConstrained total least-squareslocalisation algorithm using time difference of arrival and fre-quency difference of arrival measurements with sensor locationuncertaintiesrdquo IET Radar Sonar amp Navigation vol 6 no 9 pp891ndash899 2012

[27] F Qu and X Meng ldquoComments on rsquoConstrained total least-squares localisation algorithm using time difference of arrivaland frequency difference of arrival measurements with sensorlocation uncertaintiesrsquordquo IET Radar Sonar amp Navigation vol 8no 6 pp 692-693 2014

[28] B Hao Z Li J Si and L Guan ldquoJoint source localisation andsensor refinement using time differences of arrival and fre-quency differences of arrivalrdquo IET Signal Processing vol 8 no6 pp 588ndash600 2014

[29] K C Ho and M Sun ldquoAn accurate algebraic closed-form solu-tion for energy-based source localizationrdquo IEEETransactions onAudio Speech and Language Processing vol 15 no 8 pp 2542ndash2550 2007

[30] K C Ho and M Sun ldquoPassive source localization using timedifference of arrival and gain ratios of arrivalrdquo IEEE Transac-tions on Signal Processing vol 56 no 2 pp 464ndash477 2008

[31] B Hao Z Li J Si W Yin and Y Ren ldquoPassive multiple disjointsources localization using TDOAs and GROAs in the presenceof sensor location uncertaintiesrdquo in Proceedings of the 2012 IEEEInternational Conference on Communications ICC 2012 pp 47ndash52 Canada June 2012

[32] W H Foy ldquoPosition-location solutions by Taylorrsquos series esti-mationrdquo IEEETransactions onAerospace and Electronic Systemsvol 12 no 2 pp 187ndash194 1976

[33] K W Cheung H C So W-K Ma and Y T Chan ldquoAconstrained least squares approach to mobile positioningAlgorithms and optimalityrdquo EURASIP Journal on Applied SignalProcessing vol 2006 Article ID 20858 2006

[34] X N Lu and K C Ho ldquoAnalysis of the Degradation in SourceLocation Accuracy in the Presence of Sensor Location Errorrdquoin Proceedings of the 2006 IEEE International Conference onAcoustics Speed and Signal Processing pp 14ndash19 ToulouseFrance

[35] IMarkovsky and S VanHuffel ldquoOverview of total least-squaresmethodsrdquo Signal Processing vol 87 no 10 pp 2283ndash2302 2007

[36] T J Abatzoglou J M Mendel and G A Harada ldquoTheconstrained total least squares technique and its applications toharmonic superresolutionrdquo IEEE Transactions on Signal Pro-cessing vol 39 no 5 pp 1070ndash1087 1991

[37] K Dogancay ldquoRelationship between geometric translations andTLS estimation bias in bearings-only target localizationrdquo IEEETransactions on Signal Processing vol 56 no 3 pp 1005ndash10172008

[38] K CHo ldquoBias reduction for an explicit solution of source local-ization using TDOArdquo IEEE Transactions on Signal Processingvol 60 no 5 pp 2101ndash2114 2012

Hindawiwwwhindawicom Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Submit your manuscripts atwwwhindawicom


the nonlinear measurement equations into pseudo-linearequations except for the TS algorithm

However it must be emphasized that the accuracy ofsource location estimate may be seriously degraded by thesensor location errors regardless of the specific localizationalgorithm used In [21 34] the source location mean squareerror (MSE) is derivedwhen the sensor locations are assumedcorrect but in fact have errors Generally there exist twoclasses of methods that can mitigate the effects of theuncertainties in sensor location The first class of methods isto incorporate the statistical characteristic of sensor locationerrors into the position estimation procedure [5 6 17 21 2426 27] and the second one performs joint estimation of theunknown source locations and the inaccurate sensor posi-tions together [2 7 12 14 16 20 23 28 31] In this work weconcentrate on the latter because it can increase the accuracyof the sensor position estimates and tolerate higher noise levelbefore the thresholding effect caused by nonlinear estimationstarts to occur

It is well known that the TLS technique is an improvedleast squares (LS) method to solve an overdetermined setof linear equations Ax asymp y when there are errors not onlyin the observations y but in the coefficient matrix A aswell In [1 11 22] the TLS method is applied to sourcelocalization Note that the TLS solution can be found simplyvia the singular value decomposition (SVD) technique [35]and therefore it is computationally attractive Howeverthe TLS estimator is generally not asymptotically efficientbecause it assumes that the noise components in A and yare independent and identically distributed (iid) which israrely realistic in practical scenario The CTLS method as anatural extension of the TLS method is able to fully exploitthe noise structure in A and y [36] and hence the resultingsolution is shown to achieve the Cramer-Rao bound (CRB)Indeed the CTLS method has been successfully applied towireless location In [3] the CTLS algorithm is proposedto solve the bearing-only localization problem In [15] theCTLS localization algorithm using TDOA measurementsis presented Additionally an efficient CTLS algorithm fordetermining the position and velocity of a moving sourcebased on TDOA and FDOA measurements is developed in[25] However it is noteworthy that none of these CTLSalgorithms consider the sensor position errors which mayseriously deteriorate the positioning accuracy In order toreduce the effects of the uncertainties in sensor positions therobust CTLS localization algorithms are presented in [6 2627]

While the above-mentioned CTLS algorithms canachieve satisfactory performance it is necessary to pointout that all of them apply only to the single-source scenarioand moreover none of them can provide the joint estimationof source and sensor locations Furthermore all thealgorithm derivations and theoretical analysis are performedonly for some specific measurements thus leading tothe lack of a united framework for this problem Thispaper presents an efficient CTLS method that can locatemultiple disjoint sources and refine the erroneous sensorpositions simultaneously Different from the existingapproaches the proposed method is derived in a more

general framework that is applicable to many differentlocation measurements First a modified CTLS optimizationproblem is formulated after some algebraic manipulationsand then the corresponding Newton iterative algorithmis derived to yield the numerical solution Subsequentlyby exploiting the first-order perturbation analysis theclosed-form expression for the covariance matrix of the newCTLS estimator is deduced under the Gaussian assumptionMoreover the estimation accuracy of the CTLS solution isrigorously proved to reach the CRB before the thresholdingeffect occurs Finally we give two examples to illustrate howto utilize the proposed CTLS method for source localizationOne uses the TDOAsGROAsmeasurements and the other isbased on the TOAsFOAs parametersThe experiment resultssupport the theoretical development in this paper

The remainder of this paper is organized as follows InSection 2 the measurement model for source localization isdescribed and the problem under investigation is formulatedSection 3 derives the modified CTLS optimization modelIn Section 4 the Newton iterative algorithm is derived toprovide the joint estimation of source and sensor loca-tions Section 5 provides the closed-form expression for thecovariance matrix of the new CTLS estimator and provesits asymptotical efficiency In Section 6 two examples aregiven to illustrate how to utilize the proposed CTLS methodfor source localization Numerical simulations are presentedin Section 7 to support the theoretical development in thispaper Conclusions are drawn in Section 8 The proofs of themain results are shown in the Appendixes (available here)

2 Measurement Model andProblem Formulation

21 Nonlinear Measurement Model We consider the local-ization scenario where 119863 disjoint sources are to be locatedUnder ideal condition the location observation equationassociated with the 119889th source can be represented in a genericform as

z푑0 = f (u푑w) (1 le 119889 le 119863) (1)

where z푑0 isin R푝1times1 is the true measurement vector u푑 isinR푝2times1 is the position andor velocity vector of the 119889th sourcew isin R푝3times1 denotes the system parameter which contains thesensor positions andor velocities and f(sdot sdot) is the nonlinearfunction that depends on the specificmeasurement type used

Note that if the vectors z푑0 and w can be accuratelyobtained the localization problem is equivalent to solvinga set of nonlinear equations However z푑0 and w are notknown exactly in practice First only the noisy version of z푑0denoted as z푑 is available It can be written as

z푑 = z푑0 + n푑 = f (u푑w) + n푑 (1 le 119889 le 119863) (2)

where n푑 is the measurement noise vector that follows zero-mean Gaussian distribution with covariance matrix N푑 =119864[n푑n푇푑 ] In addition the known system parameter denotedas k is also erroneous It can be modeled as

k = w +m (3)






s (u푑w) ]] (5)




a (z0w) = B (z0w) t = B (z0w) h (uw) (6)

where





(7)



z = z0 + n


(8)

where



(9)






푗=1



(10)

where



(11)




푗=1



(12)

where





(13)


t = h (uw) = [ tw] = [h (uw)



minuwnm



m]

st [a (z k)k




m]

(15)



minuw



where

a (z k) = [a (z k)k



(17)


M (C2 (t z k))푇 M



minuwn1015840 m1015840

10038171003817100381710038171003817100381710038171003817100381710038171003817[n耠

m耠]100381710038171003817100381710038171003817100381710038171003817100381710038172

2

st [a (z k)k





sdot [ n耠m耠

](19)


[ n耠m1015840

]opt


]dagger

sdot ([a (z k)k





(20)




]dagger)푇


]dagger


]


]푇)minus1

= (Q (t z k))minus1

(21)


10038171003817100381710038171003817100381710038171003817100381710038171003817[n耠

m耠]opt

100381710038171003817100381710038171003817100381710038171003817100381710038172

2










where



120593 (uw) = [[[[[[


where


]]]]]]1205932 (uw)


]]]]]]

(26)


Ψ (uw) = [ 120597120593 (uw)120597u푇 120597120593 (uw)120597w푇 ]


]]]]


= Ψ1 (uw) +Ψ2 (uw) (27)

where



(28)


Ψ11 (uw)


]]]]]](29)

Ψ12 (uw)


]]]]]] (30)


]]]]]](31)


]]]]]] (32)



[[u(푘+1)w(푘+1)

]] = [[u(푘)w(푘)


(33)






(34)









=[[[[[[[[[[



]]]]]]]]]]= O(푝2퐷+푝3)times1

(35)



=[[[[[[[[[


푇



10038161003816100381610038161003816100381610038161003816100381610038161003816w=wctls



]]]]]]]]] (36)







] sdot [ nm]

(37)



(38)

where

H1 (u푑w) = 120597h (u푑w)120597u푇푑

= [ I푝2S1 (u푑w)]




(1 le 119889 le 119863)(39)






]푇




]




]]sdot [ n

m]

(40)

where


= (Q (t z0w))minus1


M (C2 (t z0w))푇 M

]]]]

minus1

(41)



]


]푇




]푇


] sdot [ nm]

(42)


given by

cok([uctlswctls

]) = 119864[[[120575uctls120575wctls


]푇]]= ([B (z0w) H1 (uw) B (z0w) H2 (uw)




(43)


CRB([uw]) = [[[[


]]]]

minus1

(44)





(45)




cok([uctlswctls

]) = CRB([uw]) (46)


G0 (uw)= [[[[


]]]] (47)


cok([uctlswctls

]) = [P1 P2P푇2 P3

]minus1 (48)

where




(49)


= B (f (u푑w) w) h (u푑w)(1 le 119889 le 119863)

(50)




(51)




(52)




(53)


in which

A1 (z푑0w) = 120597a (z푑0w)120597z푇푑0




(1 le 119895 le 1199013) (54)









(58)








10038171003817100381710038172 minus 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(2 le 119896 le 1198701 le 119889 le 119863) (59)






(1 le 119889 le 119863) (61)



(62)



10038171003817100381710038172 997904rArr(120588푑푘 + 1003817100381710038171003817u푑 minus w1





(63)

where


100381710038171003817100381722 (64)







(65)

where




where



s (u푑w)](68)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗2 (z푑0w)119886푗3 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]](1 le 119895 le 2)

s (u푑w) = 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(69)


a (z0w) = B (z0w) t = B (z0w) h (uw) (70)

where





(71)










120583푑푘 = 10038171003817100381710038171003817u푑푝 minus w푘푝100381710038171003817100381710038172


(1 le 119896 le 119870) (72)






(75)





u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198861푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(76)

where


(77)




u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198862푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(78)


where




where


t푑 = h (u푑w) = [[[[u푑10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝


s (u푑w)](81)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗1 (z푑0w)119886푗2 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]]

(1 le 119895 le 2)s (u푑w) = [[

10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722u푇푑Vu푑푝

]] J = O6times6퐾

(82)


a (z0w) = B (z0w) t = B (z0w) h (uw) (83)

where




(84)








2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



sitio

n RM

SE o

f the

sens

ors (

m)


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

20

205

21

215



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)









2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018













s (u푑w) ]] (5)




a (z0w) = B (z0w) t = B (z0w) h (uw) (6)

where





(7)



z = z0 + n


(8)

where



(9)






푗=1



(10)

where



(11)




푗=1



(12)

where





(13)


t = h (uw) = [ tw] = [h (uw)



minuwnm



m]

st [a (z k)k




m]

(15)



minuw



where

a (z k) = [a (z k)k



(17)


M (C2 (t z k))푇 M



minuwn1015840 m1015840

10038171003817100381710038171003817100381710038171003817100381710038171003817[n耠

m耠]100381710038171003817100381710038171003817100381710038171003817100381710038172

2

st [a (z k)k





sdot [ n耠m耠

](19)


[ n耠m1015840

]opt


]dagger

sdot ([a (z k)k





(20)




]dagger)푇


]dagger


]


]푇)minus1

= (Q (t z k))minus1

(21)


10038171003817100381710038171003817100381710038171003817100381710038171003817[n耠

m耠]opt

100381710038171003817100381710038171003817100381710038171003817100381710038172

2










where



120593 (uw) = [[[[[[


where


]]]]]]1205932 (uw)


]]]]]]

(26)


Ψ (uw) = [ 120597120593 (uw)120597u푇 120597120593 (uw)120597w푇 ]


]]]]


= Ψ1 (uw) +Ψ2 (uw) (27)

where



(28)


Ψ11 (uw)


]]]]]](29)

Ψ12 (uw)


]]]]]] (30)


]]]]]](31)


]]]]]] (32)



[[u(푘+1)w(푘+1)

]] = [[u(푘)w(푘)


(33)






(34)









=[[[[[[[[[[



]]]]]]]]]]= O(푝2퐷+푝3)times1

(35)



=[[[[[[[[[


푇



10038161003816100381610038161003816100381610038161003816100381610038161003816w=wctls



]]]]]]]]] (36)







] sdot [ nm]

(37)



(38)

where

H1 (u푑w) = 120597h (u푑w)120597u푇푑

= [ I푝2S1 (u푑w)]




(1 le 119889 le 119863)(39)






]푇




]




]]sdot [ n

m]

(40)

where


= (Q (t z0w))minus1


M (C2 (t z0w))푇 M

]]]]

minus1

(41)



]


]푇




]푇


] sdot [ nm]

(42)


given by

cok([uctlswctls

]) = 119864[[[120575uctls120575wctls


]푇]]= ([B (z0w) H1 (uw) B (z0w) H2 (uw)




(43)


CRB([uw]) = [[[[


]]]]

minus1

(44)





(45)




cok([uctlswctls

]) = CRB([uw]) (46)


G0 (uw)= [[[[


]]]] (47)


cok([uctlswctls

]) = [P1 P2P푇2 P3

]minus1 (48)

where




(49)


= B (f (u푑w) w) h (u푑w)(1 le 119889 le 119863)

(50)




(51)




(52)




(53)


in which

A1 (z푑0w) = 120597a (z푑0w)120597z푇푑0




(1 le 119895 le 1199013) (54)









(58)








10038171003817100381710038172 minus 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(2 le 119896 le 1198701 le 119889 le 119863) (59)






(1 le 119889 le 119863) (61)



(62)



10038171003817100381710038172 997904rArr(120588푑푘 + 1003817100381710038171003817u푑 minus w1





(63)

where


100381710038171003817100381722 (64)







(65)

where




where



s (u푑w)](68)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗2 (z푑0w)119886푗3 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]](1 le 119895 le 2)

s (u푑w) = 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(69)


a (z0w) = B (z0w) t = B (z0w) h (uw) (70)

where





(71)










120583푑푘 = 10038171003817100381710038171003817u푑푝 minus w푘푝100381710038171003817100381710038172


(1 le 119896 le 119870) (72)






(75)





u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198861푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(76)

where


(77)




u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198862푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(78)


where




where


t푑 = h (u푑w) = [[[[u푑10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝


s (u푑w)](81)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗1 (z푑0w)119886푗2 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]]

(1 le 119895 le 2)s (u푑w) = [[

10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722u푇푑Vu푑푝

]] J = O6times6퐾

(82)


a (z0w) = B (z0w) t = B (z0w) h (uw) (83)

where




(84)








2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

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RMSE

of t

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ourc

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)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

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180

Posit

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RMSE

of t

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sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



sitio

n RM

SE o

f the

sens

ors (

m)


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

20

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21

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2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

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RMSE

of t

he fi

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ourc

e (m

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

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200

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)









2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018











푗=1



(12)

where





(13)


t = h (uw) = [ tw] = [h (uw)



minuwnm



m]

st [a (z k)k




m]

(15)



minuw



where

a (z k) = [a (z k)k



(17)


M (C2 (t z k))푇 M



minuwn1015840 m1015840

10038171003817100381710038171003817100381710038171003817100381710038171003817[n耠

m耠]100381710038171003817100381710038171003817100381710038171003817100381710038172

2

st [a (z k)k





sdot [ n耠m耠

](19)


[ n耠m1015840

]opt


]dagger

sdot ([a (z k)k





(20)




]dagger)푇


]dagger


]


]푇)minus1

= (Q (t z k))minus1

(21)


10038171003817100381710038171003817100381710038171003817100381710038171003817[n耠

m耠]opt

100381710038171003817100381710038171003817100381710038171003817100381710038172

2










where



120593 (uw) = [[[[[[


where


]]]]]]1205932 (uw)


]]]]]]

(26)


Ψ (uw) = [ 120597120593 (uw)120597u푇 120597120593 (uw)120597w푇 ]


]]]]


= Ψ1 (uw) +Ψ2 (uw) (27)

where



(28)


Ψ11 (uw)


]]]]]](29)

Ψ12 (uw)


]]]]]] (30)


]]]]]](31)


]]]]]] (32)



[[u(푘+1)w(푘+1)

]] = [[u(푘)w(푘)


(33)






(34)









=[[[[[[[[[[



]]]]]]]]]]= O(푝2퐷+푝3)times1

(35)



=[[[[[[[[[


푇



10038161003816100381610038161003816100381610038161003816100381610038161003816w=wctls



]]]]]]]]] (36)







] sdot [ nm]

(37)



(38)

where

H1 (u푑w) = 120597h (u푑w)120597u푇푑

= [ I푝2S1 (u푑w)]




(1 le 119889 le 119863)(39)






]푇




]




]]sdot [ n

m]

(40)

where


= (Q (t z0w))minus1


M (C2 (t z0w))푇 M

]]]]

minus1

(41)



]


]푇




]푇


] sdot [ nm]

(42)


given by

cok([uctlswctls

]) = 119864[[[120575uctls120575wctls


]푇]]= ([B (z0w) H1 (uw) B (z0w) H2 (uw)




(43)


CRB([uw]) = [[[[


]]]]

minus1

(44)





(45)




cok([uctlswctls

]) = CRB([uw]) (46)


G0 (uw)= [[[[


]]]] (47)


cok([uctlswctls

]) = [P1 P2P푇2 P3

]minus1 (48)

where




(49)


= B (f (u푑w) w) h (u푑w)(1 le 119889 le 119863)

(50)




(51)




(52)




(53)


in which

A1 (z푑0w) = 120597a (z푑0w)120597z푇푑0




(1 le 119895 le 1199013) (54)









(58)








10038171003817100381710038172 minus 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(2 le 119896 le 1198701 le 119889 le 119863) (59)






(1 le 119889 le 119863) (61)



(62)



10038171003817100381710038172 997904rArr(120588푑푘 + 1003817100381710038171003817u푑 minus w1





(63)

where


100381710038171003817100381722 (64)







(65)

where




where



s (u푑w)](68)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗2 (z푑0w)119886푗3 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]](1 le 119895 le 2)

s (u푑w) = 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(69)


a (z0w) = B (z0w) t = B (z0w) h (uw) (70)

where





(71)










120583푑푘 = 10038171003817100381710038171003817u푑푝 minus w푘푝100381710038171003817100381710038172


(1 le 119896 le 119870) (72)






(75)





u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198861푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(76)

where


(77)




u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198862푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(78)


where




where


t푑 = h (u푑w) = [[[[u푑10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝


s (u푑w)](81)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗1 (z푑0w)119886푗2 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]]

(1 le 119895 le 2)s (u푑w) = [[

10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722u푇푑Vu푑푝

]] J = O6times6퐾

(82)


a (z0w) = B (z0w) t = B (z0w) h (uw) (83)

where




(84)








2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



sitio

n RM

SE o

f the

sens

ors (

m)


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

20

205

21

215



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)









2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018









sdot [ n耠m耠

](19)


[ n耠m1015840

]opt


]dagger

sdot ([a (z k)k





(20)




]dagger)푇


]dagger


]


]푇)minus1

= (Q (t z k))minus1

(21)


10038171003817100381710038171003817100381710038171003817100381710038171003817[n耠

m耠]opt

100381710038171003817100381710038171003817100381710038171003817100381710038172

2










where



120593 (uw) = [[[[[[


where


]]]]]]1205932 (uw)


]]]]]]

(26)


Ψ (uw) = [ 120597120593 (uw)120597u푇 120597120593 (uw)120597w푇 ]


]]]]


= Ψ1 (uw) +Ψ2 (uw) (27)

where



(28)


Ψ11 (uw)


]]]]]](29)

Ψ12 (uw)


]]]]]] (30)


]]]]]](31)


]]]]]] (32)



[[u(푘+1)w(푘+1)

]] = [[u(푘)w(푘)


(33)






(34)









=[[[[[[[[[[



]]]]]]]]]]= O(푝2퐷+푝3)times1

(35)



=[[[[[[[[[


푇



10038161003816100381610038161003816100381610038161003816100381610038161003816w=wctls



]]]]]]]]] (36)







] sdot [ nm]

(37)



(38)

where

H1 (u푑w) = 120597h (u푑w)120597u푇푑

= [ I푝2S1 (u푑w)]




(1 le 119889 le 119863)(39)






]푇




]




]]sdot [ n

m]

(40)

where


= (Q (t z0w))minus1


M (C2 (t z0w))푇 M

]]]]

minus1

(41)



]


]푇




]푇


] sdot [ nm]

(42)


given by

cok([uctlswctls

]) = 119864[[[120575uctls120575wctls


]푇]]= ([B (z0w) H1 (uw) B (z0w) H2 (uw)




(43)


CRB([uw]) = [[[[


]]]]

minus1

(44)





(45)




cok([uctlswctls

]) = CRB([uw]) (46)


G0 (uw)= [[[[


]]]] (47)


cok([uctlswctls

]) = [P1 P2P푇2 P3

]minus1 (48)

where




(49)


= B (f (u푑w) w) h (u푑w)(1 le 119889 le 119863)

(50)




(51)




(52)




(53)


in which

A1 (z푑0w) = 120597a (z푑0w)120597z푇푑0




(1 le 119895 le 1199013) (54)









(58)








10038171003817100381710038172 minus 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(2 le 119896 le 1198701 le 119889 le 119863) (59)






(1 le 119889 le 119863) (61)



(62)



10038171003817100381710038172 997904rArr(120588푑푘 + 1003817100381710038171003817u푑 minus w1





(63)

where


100381710038171003817100381722 (64)







(65)

where




where



s (u푑w)](68)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗2 (z푑0w)119886푗3 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]](1 le 119895 le 2)

s (u푑w) = 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(69)


a (z0w) = B (z0w) t = B (z0w) h (uw) (70)

where





(71)










120583푑푘 = 10038171003817100381710038171003817u푑푝 minus w푘푝100381710038171003817100381710038172


(1 le 119896 le 119870) (72)






(75)





u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198861푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(76)

where


(77)




u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198862푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(78)


where




where


t푑 = h (u푑w) = [[[[u푑10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝


s (u푑w)](81)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗1 (z푑0w)119886푗2 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]]

(1 le 119895 le 2)s (u푑w) = [[

10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722u푇푑Vu푑푝

]] J = O6times6퐾

(82)


a (z0w) = B (z0w) t = B (z0w) h (uw) (83)

where




(84)








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RMSE

of t

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ourc

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)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

Posit

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RMSE

of t

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sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



sitio

n RM

SE o

f the

sens

ors (

m)


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

20

205

21

215



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)









2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018









= Ψ1 (uw) +Ψ2 (uw) (27)

where



(28)


Ψ11 (uw)


]]]]]](29)

Ψ12 (uw)


]]]]]] (30)


]]]]]](31)


]]]]]] (32)



[[u(푘+1)w(푘+1)

]] = [[u(푘)w(푘)


(33)






(34)









=[[[[[[[[[[



]]]]]]]]]]= O(푝2퐷+푝3)times1

(35)



=[[[[[[[[[


푇



10038161003816100381610038161003816100381610038161003816100381610038161003816w=wctls



]]]]]]]]] (36)







] sdot [ nm]

(37)



(38)

where

H1 (u푑w) = 120597h (u푑w)120597u푇푑

= [ I푝2S1 (u푑w)]




(1 le 119889 le 119863)(39)






]푇




]




]]sdot [ n

m]

(40)

where


= (Q (t z0w))minus1


M (C2 (t z0w))푇 M

]]]]

minus1

(41)



]


]푇




]푇


] sdot [ nm]

(42)


given by

cok([uctlswctls

]) = 119864[[[120575uctls120575wctls


]푇]]= ([B (z0w) H1 (uw) B (z0w) H2 (uw)




(43)


CRB([uw]) = [[[[


]]]]

minus1

(44)





(45)




cok([uctlswctls

]) = CRB([uw]) (46)


G0 (uw)= [[[[


]]]] (47)


cok([uctlswctls

]) = [P1 P2P푇2 P3

]minus1 (48)

where




(49)


= B (f (u푑w) w) h (u푑w)(1 le 119889 le 119863)

(50)




(51)




(52)




(53)


in which

A1 (z푑0w) = 120597a (z푑0w)120597z푇푑0




(1 le 119895 le 1199013) (54)









(58)








10038171003817100381710038172 minus 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(2 le 119896 le 1198701 le 119889 le 119863) (59)






(1 le 119889 le 119863) (61)



(62)



10038171003817100381710038172 997904rArr(120588푑푘 + 1003817100381710038171003817u푑 minus w1





(63)

where


100381710038171003817100381722 (64)







(65)

where




where



s (u푑w)](68)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗2 (z푑0w)119886푗3 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]](1 le 119895 le 2)

s (u푑w) = 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(69)


a (z0w) = B (z0w) t = B (z0w) h (uw) (70)

where





(71)










120583푑푘 = 10038171003817100381710038171003817u푑푝 minus w푘푝100381710038171003817100381710038172


(1 le 119896 le 119870) (72)






(75)





u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198861푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(76)

where


(77)




u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198862푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(78)


where




where


t푑 = h (u푑w) = [[[[u푑10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝


s (u푑w)](81)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗1 (z푑0w)119886푗2 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]]

(1 le 119895 le 2)s (u푑w) = [[

10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722u푇푑Vu푑푝

]] J = O6times6퐾

(82)


a (z0w) = B (z0w) t = B (z0w) h (uw) (83)

where




(84)








2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



sitio

n RM

SE o

f the

sens

ors (

m)


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

20

205

21

215



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)









2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018
















=[[[[[[[[[[



]]]]]]]]]]= O(푝2퐷+푝3)times1

(35)



=[[[[[[[[[


푇



10038161003816100381610038161003816100381610038161003816100381610038161003816w=wctls



]]]]]]]]] (36)







] sdot [ nm]

(37)



(38)

where

H1 (u푑w) = 120597h (u푑w)120597u푇푑

= [ I푝2S1 (u푑w)]




(1 le 119889 le 119863)(39)






]푇




]




]]sdot [ n

m]

(40)

where


= (Q (t z0w))minus1


M (C2 (t z0w))푇 M

]]]]

minus1

(41)



]


]푇




]푇


] sdot [ nm]

(42)


given by

cok([uctlswctls

]) = 119864[[[120575uctls120575wctls


]푇]]= ([B (z0w) H1 (uw) B (z0w) H2 (uw)




(43)


CRB([uw]) = [[[[


]]]]

minus1

(44)





(45)




cok([uctlswctls

]) = CRB([uw]) (46)


G0 (uw)= [[[[


]]]] (47)


cok([uctlswctls

]) = [P1 P2P푇2 P3

]minus1 (48)

where




(49)


= B (f (u푑w) w) h (u푑w)(1 le 119889 le 119863)

(50)




(51)




(52)




(53)


in which

A1 (z푑0w) = 120597a (z푑0w)120597z푇푑0




(1 le 119895 le 1199013) (54)









(58)








10038171003817100381710038172 minus 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(2 le 119896 le 1198701 le 119889 le 119863) (59)






(1 le 119889 le 119863) (61)



(62)



10038171003817100381710038172 997904rArr(120588푑푘 + 1003817100381710038171003817u푑 minus w1





(63)

where


100381710038171003817100381722 (64)







(65)

where




where



s (u푑w)](68)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗2 (z푑0w)119886푗3 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]](1 le 119895 le 2)

s (u푑w) = 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(69)


a (z0w) = B (z0w) t = B (z0w) h (uw) (70)

where





(71)










120583푑푘 = 10038171003817100381710038171003817u푑푝 minus w푘푝100381710038171003817100381710038172


(1 le 119896 le 119870) (72)






(75)





u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198861푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(76)

where


(77)




u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198862푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(78)


where




where


t푑 = h (u푑w) = [[[[u푑10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝


s (u푑w)](81)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗1 (z푑0w)119886푗2 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]]

(1 le 119895 le 2)s (u푑w) = [[

10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722u푇푑Vu푑푝

]] J = O6times6퐾

(82)


a (z0w) = B (z0w) t = B (z0w) h (uw) (83)

where




(84)








2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



sitio

n RM

SE o

f the

sens

ors (

m)


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

20

205

21

215



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)









2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018











]푇




]




]]sdot [ n

m]

(40)

where


= (Q (t z0w))minus1


M (C2 (t z0w))푇 M

]]]]

minus1

(41)



]


]푇




]푇


] sdot [ nm]

(42)


given by

cok([uctlswctls

]) = 119864[[[120575uctls120575wctls


]푇]]= ([B (z0w) H1 (uw) B (z0w) H2 (uw)




(43)


CRB([uw]) = [[[[


]]]]

minus1

(44)





(45)




cok([uctlswctls

]) = CRB([uw]) (46)


G0 (uw)= [[[[


]]]] (47)


cok([uctlswctls

]) = [P1 P2P푇2 P3

]minus1 (48)

where




(49)


= B (f (u푑w) w) h (u푑w)(1 le 119889 le 119863)

(50)




(51)




(52)




(53)


in which

A1 (z푑0w) = 120597a (z푑0w)120597z푇푑0




(1 le 119895 le 1199013) (54)









(58)








10038171003817100381710038172 minus 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(2 le 119896 le 1198701 le 119889 le 119863) (59)






(1 le 119889 le 119863) (61)



(62)



10038171003817100381710038172 997904rArr(120588푑푘 + 1003817100381710038171003817u푑 minus w1





(63)

where


100381710038171003817100381722 (64)







(65)

where




where



s (u푑w)](68)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗2 (z푑0w)119886푗3 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]](1 le 119895 le 2)

s (u푑w) = 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(69)


a (z0w) = B (z0w) t = B (z0w) h (uw) (70)

where





(71)










120583푑푘 = 10038171003817100381710038171003817u푑푝 minus w푘푝100381710038171003817100381710038172


(1 le 119896 le 119870) (72)






(75)





u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198861푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(76)

where


(77)




u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198862푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(78)


where




where


t푑 = h (u푑w) = [[[[u푑10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝


s (u푑w)](81)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗1 (z푑0w)119886푗2 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]]

(1 le 119895 le 2)s (u푑w) = [[

10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722u푇푑Vu푑푝

]] J = O6times6퐾

(82)


a (z0w) = B (z0w) t = B (z0w) h (uw) (83)

where




(84)








2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



sitio

n RM

SE o

f the

sens

ors (

m)


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

20

205

21

215



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)









2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018










(45)




cok([uctlswctls

]) = CRB([uw]) (46)


G0 (uw)= [[[[


]]]] (47)


cok([uctlswctls

]) = [P1 P2P푇2 P3

]minus1 (48)

where




(49)


= B (f (u푑w) w) h (u푑w)(1 le 119889 le 119863)

(50)




(51)




(52)




(53)


in which

A1 (z푑0w) = 120597a (z푑0w)120597z푇푑0




(1 le 119895 le 1199013) (54)









(58)








10038171003817100381710038172 minus 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(2 le 119896 le 1198701 le 119889 le 119863) (59)






(1 le 119889 le 119863) (61)



(62)



10038171003817100381710038172 997904rArr(120588푑푘 + 1003817100381710038171003817u푑 minus w1





(63)

where


100381710038171003817100381722 (64)







(65)

where




where



s (u푑w)](68)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗2 (z푑0w)119886푗3 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]](1 le 119895 le 2)

s (u푑w) = 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(69)


a (z0w) = B (z0w) t = B (z0w) h (uw) (70)

where





(71)










120583푑푘 = 10038171003817100381710038171003817u푑푝 minus w푘푝100381710038171003817100381710038172


(1 le 119896 le 119870) (72)






(75)





u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198861푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(76)

where


(77)




u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198862푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(78)


where




where


t푑 = h (u푑w) = [[[[u푑10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝


s (u푑w)](81)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗1 (z푑0w)119886푗2 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]]

(1 le 119895 le 2)s (u푑w) = [[

10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722u푇푑Vu푑푝

]] J = O6times6퐾

(82)


a (z0w) = B (z0w) t = B (z0w) h (uw) (83)

where




(84)








2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



sitio

n RM

SE o

f the

sens

ors (

m)


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

20

205

21

215



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)









2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018









in which

A1 (z푑0w) = 120597a (z푑0w)120597z푇푑0




(1 le 119895 le 1199013) (54)









(58)








10038171003817100381710038172 minus 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(2 le 119896 le 1198701 le 119889 le 119863) (59)






(1 le 119889 le 119863) (61)



(62)



10038171003817100381710038172 997904rArr(120588푑푘 + 1003817100381710038171003817u푑 minus w1





(63)

where


100381710038171003817100381722 (64)







(65)

where




where



s (u푑w)](68)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗2 (z푑0w)119886푗3 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]](1 le 119895 le 2)

s (u푑w) = 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(69)


a (z0w) = B (z0w) t = B (z0w) h (uw) (70)

where





(71)










120583푑푘 = 10038171003817100381710038171003817u푑푝 minus w푘푝100381710038171003817100381710038172


(1 le 119896 le 119870) (72)






(75)





u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198861푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(76)

where


(77)




u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198862푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(78)


where




where


t푑 = h (u푑w) = [[[[u푑10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝


s (u푑w)](81)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗1 (z푑0w)119886푗2 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]]

(1 le 119895 le 2)s (u푑w) = [[

10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722u푇푑Vu푑푝

]] J = O6times6퐾

(82)


a (z0w) = B (z0w) t = B (z0w) h (uw) (83)

where




(84)








2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



sitio

n RM

SE o

f the

sens

ors (

m)


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

20

205

21

215



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)









2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018











(1 le 119889 le 119863) (61)



(62)



10038171003817100381710038172 997904rArr(120588푑푘 + 1003817100381710038171003817u푑 minus w1





(63)

where


100381710038171003817100381722 (64)







(65)

where




where



s (u푑w)](68)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗2 (z푑0w)119886푗3 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]](1 le 119895 le 2)

s (u푑w) = 1003817100381710038171003817u푑 minus w110038171003817100381710038172


(69)


a (z0w) = B (z0w) t = B (z0w) h (uw) (70)

where





(71)










120583푑푘 = 10038171003817100381710038171003817u푑푝 minus w푘푝100381710038171003817100381710038172


(1 le 119896 le 119870) (72)






(75)





u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198861푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(76)

where


(77)




u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198862푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(78)


where




where


t푑 = h (u푑w) = [[[[u푑10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝


s (u푑w)](81)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗1 (z푑0w)119886푗2 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]]

(1 le 119895 le 2)s (u푑w) = [[

10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722u푇푑Vu푑푝

]] J = O6times6퐾

(82)


a (z0w) = B (z0w) t = B (z0w) h (uw) (83)

where




(84)








2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



sitio

n RM

SE o

f the

sens

ors (

m)


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

20

205

21

215



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)









2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018

















120583푑푘 = 10038171003817100381710038171003817u푑푝 minus w푘푝100381710038171003817100381710038172


(1 le 119896 le 119870) (72)






(75)





u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198861푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(76)

where


(77)




u푑푝u푑V10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝

]]]]]]]= 1198862푘 (z푑0w)

(1 le 119896 le 119870 1 le 119889 le 119863)

(78)


where




where


t푑 = h (u푑w) = [[[[u푑10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝


s (u푑w)](81)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗1 (z푑0w)119886푗2 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]]

(1 le 119895 le 2)s (u푑w) = [[

10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722u푇푑Vu푑푝

]] J = O6times6퐾

(82)


a (z0w) = B (z0w) t = B (z0w) h (uw) (83)

where




(84)








2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



sitio

n RM

SE o

f the

sens

ors (

m)


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

20

205

21

215



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)









2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018









where




where


t푑 = h (u푑w) = [[[[u푑10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722

u푇푑Vu푑푝


s (u푑w)](81)

in which

B푗 (z푑0w) =[[[[[[[[


]]]]]]]]

a푗 (z푑0w) =[[[[[[[[

119886푗1 (z푑0w)119886푗2 (z푑0w)119886푗퐾 (z푑0w)

]]]]]]]]

(1 le 119895 le 2)s (u푑w) = [[

10038171003817100381710038171003817u푑푝1003817100381710038171003817100381722u푇푑Vu푑푝

]] J = O6times6퐾

(82)


a (z0w) = B (z0w) t = B (z0w) h (uw) (83)

where




(84)








2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



sitio

n RM

SE o

f the

sens

ors (

m)


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

20

205

21

215



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)









2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018










2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



sitio

n RM

SE o

f the

sens

ors (

m)


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

20

205

21

215



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)









2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018









sitio

n RM

SE o

f the

sens

ors (

m)


2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

20

205

21

215



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he se

cond

sour

ce (m



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

180

200

Posit

ion

RMSE

of t

he th

ird so

urce

(m)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)









2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018










2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

0

10

20

30

40

50

60

70

80

90

100

Nor

m o

f sou

rce p

ositi

on b

ias (

m)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

10

20

30

40

50

60

Nor

m o

f sou

rce p

ositi

on b

ias (

m)









2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018












2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

40

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

16

17

18

19

2

21

22

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)






2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018










2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

50

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

18

185

19

195

2

205

21

215

22

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

70

80

90

100

110

120

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

2

205

21

215

22

225

23

235

24

Velo

city

RM

SE o

f the

third

sour

ce (m

s)





2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018










2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

60

65

70

75

80

85

90Po

sitio

n RM

SE o

f the

sens

ors (

m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2011

19

195

2

205

21

215

Velo

city

RM

SE o

f the

sens

ors (

ms

)





2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he fi

rst s

ourc

e (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

firs

t sou

rce (

ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018










2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

50

100

150

Posit

ion

RMSE

of t

he se

cond

sour

ce (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

seco

nd so

urce

(ms

)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

160

Posit

ion

RMSE

of t

he th

ird so

urce

(m)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

1

2

3

4

5

6

Velo

city

RM

SE o

f the

third

sour

ce (m

s)




2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018










2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

20

40

60

80

100

120

140

Posit

ion

RMSE

of t

he se

nsor

s (m

)



2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012

0

05

1

15

2

25

3

35

4

45

5

Velo

city

RM

SE o

f the

sens

ors (

ms

)







(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

5

10

15

20

25

30

35

40

Nor

m o

f sou

rce p

ositi

on b

ias (

m)






(the first source)

(the second source)

(the third source)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201

0

01

02

03

04

05

06

Nor

m o

f sou

rce v

eloci

ty b

ias (

ms

)






Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018












Acknowledgments




References















































Journal of












Journal of







Volume 2018






Volume 2018































Journal of












Journal of







Volume 2018






Volume 2018








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